Research ArticleCharged System Search Algorithm Utilized forStructural Damage Detection

Zahra Tabrizian1 Gholamreza Ghodrati Amiri2 and Morteza Hossein Ali Beigy1

1 College of Civil Engineering Babol Noshirvani University of Technology Babol Iran2 Centre of Excellence for Fundamental Studies in Structural Engineering Iran University of Science and TechnologyNarmak Tehran Iran

Correspondence should be addressed to Gholamreza Ghodrati Amiri ghodratiiustacir

Received 12 June 2013 Revised 8 April 2014 Accepted 9 April 2014 Published 20 May 2014

Academic Editor Gyuhae Park

Copyright copy 2014 Zahra Tabrizian et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents damage detection and assessment methodology based on the changes in dynamic parameters of a structuralsystemThemethod is applied at an element level using a finite elementmodel According to continuumdamagemechanics damageis represented by a reduction factor of the element stiffness A recently developed metaheuristic optimization algorithm known asthe charged system search (CSS) is utilized for locating and quantifying the damaged areas of the structure In order to demonstratethe abilities of this method three examples are included comprising of a 10-elements cantilever beam a Bowstring plane truss anda 39-element three-story three-bay plane frame The possible damage types in structures by considering several damage scenariosand using incomplete modal data are modeled Finally results are obtained from the CSS algorithm by detecting damage in thesestructures and compared to the results of the PSOPC algorithm In addition the effect of noise is shown in the results of the CSSalgorithm by suitable diagrams As is illustrated this method has acceptable results in the structural detection damage with lowcomputational time

1 Introduction

Many civil engineering structures like other structuresdesigned with older codes have been suffering damage anddeterioration in recent years which caused reduction intheir performance Due to this effect damage assessment ofthese structures is becoming increasingly essential in order todetermine their safety and reliabilityThemethodmentionedin this paper is either visual or nondestructive which leadus to the best answers Considering global damage detectionmethods applicable to complex structures methods basedon modal testing Ewins 1984 [1] and signal processing ismade to identify damage in civil engineering and mechanicsThese methods study changes in the dynamic characteristicsof the structure such as natural frequencies andmode shapeswhich were studied by Doebling et al 1998 [2] and Huet al 2001 [3] The undamaged and damaged structurecompare with the identification of the location and theseverity of damage Natural frequencies and mode shapes aredamage indicators which are used as the dynamic parameters

Methods based on the measurement of natural frequenciesare very simple since this parameter can be determinedby measuring at only one point of the structure (Salawu1997 [4] and Bicanic and Chen 1997 [5]) Some derivativesof mode shapes such as mode shape curvatures are moresensitive to small perturbations than modal displacementsand therefore can also be used to detect damage (Pandey etal 1991 [6]) However their applicability is minimal sincetheir estimation from experimental data is very difficult andtherefore they are very uncertain from a statistical point ofview Mottershead and Friswell 1993 have studied modelupdating methods In addition to the updating methodsbased on traditional optimization techniques new methodshave been developed in inverse procedures which are calledevolutionary algorithms [7]

In many structures some local damagemay occur duringtheir functional age In order to develop the efficiency of thestructures it is necessary to properly identify the damageplaces and their severity and then repair them There are

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 194753 13 pageshttpdxdoiorg1011552014194753

2 Shock and Vibration

many methods that have been introduced to correctly findout the place and severity of structural damage

Perera and Torres 2006 studied a nondestructive globaldamage detection and assessment methodology based onthe changes in frequencies and mode shapes of vibrationof a structural system It was shown that the proposed GAyields a suitable damage places and severity detection fromtraditional methods [8]

A two-stage method for determining place and severityof multiple structural damage by combining the adaptiveneurofuzzy inference system (ANFIS) and particle swarmoptimization (PSO) has been proposed by Fallahian andSeyedpoor [9] In a studyKoh andDyke [10] have determinedthe place and severity of multiple damage by iterativelysearching for a combination of structural responses thatmaximizes a correlation coefficient named the multipledamage location assurance criterion (MDLAC) via a geneticalgorithm (GA) Damage detection by a hybrid techniqueincluding a real-parameter genetic algorithm and grey rela-tion analysis has been presented by He and Hwang [11]They used first a grey relation analysis to exclude impossibledamage places such that the number of design variables couldbe reduced Second a real-parameter genetic algorithm wascombined with simulated annealing for finding the actualdamage The damage identification of a beam-like structurehas been formulated as an optimization problem and a GAhas been employed to find the damage place and severityby minimizing the cost function which is based on thedifference of measured and calculated natural frequencies Atwo-stage method for determining the place and severity ofmultiple structural damage by using an information fusiontechnique and a GA has been presented by Guo and Li [12]In the first stage the damage is localized by using evidencetheory and then a microsearch genetic algorithm (MSGA)has been planned to determine the damage size Yu and Chendetermined the place and severity of damage in a bridgeFirst two objective functions are defined One is defined asminimizing the sum of differences between the modal databefore and after damage in traditional wayThe other is newlydefined based on modal flexibility which is combined withanother function able to predict damage location Secondlyan improved particle swarm optimization (PSO) algorithmis developed based on the macroeconomic strategies andused to solve the multiple-objective optimization problemon bridge damage identification The results show that theprocedure is very promising for locating and quantifyingdamaged elements of bridge structures and considerablyimproves predictions based on the modal flexibility as wellas the PSO method [13] An approach for detecting damagein structural members based on continuum damage modelusing an algorithmnamedBig Bang-Big Crunch (BB-BC) haspresented suitable accuracy in results [14]

In recent years the use of metaheuristic methods hasincreased and great efforts to increase the power of algo-rithms (Efficiency) and reduce optimization time (moreconvergence speed) have been done Recently a new androbust algorithm was presented by Kaveh and Talatahari socalled charged system search [15] This algorithm is based onsome laws from electrostatics and the Newtonian mechanics

The charged system search (CSS) utilizes a number of chargedparticles (CP) which affect each other based on their fitnessvalues and separation distances consideringCoulombGaussandNewtonian lawsThe resultant forces and themotion lawsdetermine the new location of the particles The harmonysearch-based handling approach is utilized for controlling thevariable constraint

In this paper the CSS algorithm is utilized to predict thedamage place and severity for different types of structuresThe structure is modeled with the finite element method andthe damage identification is performed at element level withincomplete modal data

In order to show the power of the CSS algorithm in struc-tural damage detection the obtained results are comparedto the results of the PSOPC algorithm [16] PSOPC algo-rithm is a particle swarm optimization method with passivecongregationrsquos capacity [17] High abilities of this algorithmwith acceptable results in structural damage detection areillustrated [16]

The results show that the proposed method is capableof detecting the location and severity of diagnosis evenin large-scale structures with a large number of damagedelements it achieved acceptable resultsThe effect of the noiseis considered by assigning the noise in natural frequencieson the results of the CSS algorithm The graphs indicate thatthe noise in input data may reduce the accuracy of damagedetection

2 Theory

Existing Structural damage detection techniques can beclassified into two major categories the dynamic and staticmethods [18] Both methods are based on the finite elementutilizing experimental test data requiring dynamic and statictest data correspondingly In addition dynamic methodshave shown their advantages in comparison with static onesThe natural frequencies of a structure can be considered tobe valuable with the dynamic data Determining the levelof correlation between the calculated and predicted naturalfrequencies can provide a simple tool for finding the place andseverity of structural damage [19] Structural damage detec-tion is calculated with changes in structural characteristicsand it is possible with global evaluating Doebling et al havediscussed vibration-based techniques in a literature review[20] In this section the construction of dynamics of damagedstructures is discussed

The parameter vector used for evaluating the correlationcoefficients (the ratio of the changes first 119899119891 natural fre-quency Δ119865 due to structural damage) is

Δ119865 =

119865119873 minus 119865119864

119865119873

(1)

where 119865119873 and 119865119864 denote the natural frequency vectors of theundamaged and damaged structure Similarly the parametervector predicted from an analytical model can be definedcorrespondingly as

120575119865 (119883) =

119865119873 minus 119865 (119883)

119865119873

(2)

Shock and Vibration 3

where 119865(119883) is a natural frequency vector that can be pre-dicted from an analytic model and 119883 = [119909

1 1199092 119909

119899]

represents a damage variable vector containing the damageseverity of all 119899 structural elements Given a pair of param-eter vectors one can estimate the level of correlation inseveral ways An efficient way is to evaluate a correlation-based index termed themultiples damage location assurancecriterion (MDLAC) and covered in the following form [19]

MDLAC (119883) =10038161003816100381610038161003816Δ119865119879sdot 120575119865 (119883)

10038161003816100381610038161003816

2

(Δ119865119879sdot Δ119865) (120575119865

119879(119883) sdot 120575119865 (119883))

0 lt MDLAC lt 1

(3)

Two frequency change vectors are compared with MDLACone calculated from the structural tests and the other froma structural model analysis When the vector of analyticalfrequencies becomes the same as the frequency vector ofthe damaged structure MDLAC will be maximal That is119865(119883) = 119865 so considering this theory can be used to finda set of damage variables maximizing the MDLAC using anoptimization algorithm

Find 119883 = [119909119894 1199092 119909119899]

Maximize 119908 (119883) = MDLAC (119883) (4)

The damage severity can take values only from the set thatis given from [ 0 1 ] a set of continuous values Moreoverthe objective function that should be maximized is 119908 Asmentioned the damage occurrence in a structural elementdecreases the element stiffness Thus one of the methods forthe damage identification problem is simulation damage bydecreasing one of the stiffness parameters of the element suchas the modulus of elasticity (119864) cross-sectional area (119860) andinertia moment (119868) In this study the damage variables aredefined via a relative reduction of the elasticity modulus ofan element as

119870119889119894 = (1 minus 119909119894)119870

ℎ119894

(5)

119870119889119894 is the stiffness matrix of damaged element 119894 and119870ℎ119894 is the

stiffness matrix of healthy element 119894119864 is the primary modulus of elasticity and 119864119894 is the final

modulus of elasticity of the 119894th element The MDLAC as anobjective function for the optimization algorithm is moresensitive to damaged elements than undamaged elementsIt means that this method can find the true place of thedamaged elements but it may find an undamaged element asa damaged one Therefore in this study a new function andnew optimization algorithm is discussed the new function ispresented as [21]

obj (119883) = 1

119899119891

119899119891

sum

119894=1

min (119891119909119894 119891119864119894)max (119891119909119894 119891119864119894)

(6)

where119891119909119894119891119864119894 are the 119894th components of vectors 119865(119883) and 119865119864correspondingly

The obj(119883) function can rapidly find the locations ofhealthy elements when compared to theMDLAC however it

is very probable that it finds a damaged element as a healthyone Therefore in this study a combinational function of(3) and (6) called here the efficient correlation-based index(ECBI) is used as [21]

ECBI (119883) = 1

2

(MDLAC (119883) + obj (119883)) (7)

3 Charged System Search Algorithm

A new type of metaheuristic algorithms is introduced byKaveh and Talatahari [15] which is called charge systemsearch The charged system search (CSS) is based onCoulomb and Gauss laws from electrical physics and thegoverning laws of motion from the Newtonian mechanicsIn this algorithm each agent is a charged particle (CP) EachCP is considered as a charged sphere which exerts an electricforce on other CPs according to Coulomb and Gauss lawsThe resultant forces and the motion laws determine the newlocation of the CPs [22] The new positions of the chargedparticles in the first iteration are determined randomly andfor next iterations are obtained as follows

119883119895new = rand1198951 sdot 119896119886 sdot119865119895

119898119895

sdot Δ1199052

+ rand1198952 sdot 119896V sdot 119881119895old sdot Δ119905 + 119883119895old

119881119895new =119883119895new minus 119883119895old

Δ119905

(8)

where 119870119886 and 119870V are the acceleration and the velocitycoefficients respectively rand1198951 and rand1198952 are two randomuniformly distributed in the range (0 1) and the resultantforces vector for 119895th CP 119865119895 is calculated as

119865119895 = 119902119894 sum

119894119894 = 119895

(

119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119901119894119895 (119883119894 minus 119883119895)

119895 = 1 2 119873

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(9)

where the magnitude of charge for each CP 119902119894 is defined as

119902119894 =fit (119894) minus fitworstfitbest minus fitworst

119894 = 1 2 119873 (10)

where fitbest and fitworst are the best and the worst fitness ofall the CPs fit(119894) is the fitness of the agent 119894 and119873 is the totalnumber of charged particlesThe separation distance betweentwo CPs 119903119894119895 is obtained as follows

119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

10038171003817100381710038171003817

10038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576

(11)

where119883119894 and119883119895 are the positions of the 119894th and 119895th CPs119883bestis the position of the best current CP and 120576 is a small positive

4 Shock and Vibration

Search

No

Yes

No

Yes

Terminationcriterion

Step 1 Step 2

Step 3

Step 4

Step 1 Step 2

Step 3

Initialize specification of problem and algorithm

parameters and determinethe initial charged particles

Find the values of the objective function for the

CPs and rank of CPs in an increasing order

Store some of the best CPs in

the charged memory (CM)

Determine the probability of

moving and force vector for each CP

Determine the new positions and

velocities of the CPs

Correct the CP positions using an algorithm based on HS if the CP does not satisfy

the side limits

Evaluate the objective function and rank the

CPs according to their quality

Include the better vectors in the CM and exclude the worst ones

from the CM

Stop

Are the new CPs better than the stored ones

in CM

Initialization

Figure 1 Flowchart of the CSS algorithm

1 2 96 8 1043 5 7

1007mW12 times 65

Figure 2 A cantilever beam geometry

0 500 1000 1500 2000 2500 30000

005

01

015

02

025

Number of analyses

Obj

ectiv

e fun

ctio

n

CSSPSOPC

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 30000

005

01

015

02

025

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 3 The convergence history of the 10-element beam for the CSS and PSOPC algorithms

Shock and Vibration 5

0 1 2 3 4 5 6 7 8 9 100

5

10

15

Element number

Dam

age (

)

Incomplete noisy dataComplete noisy dataActual damage

Figure 4 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 1

numberThe probability of moving each CP toward the otherCPs is determined using the following function

119901119894119895 =

1

fit (119894) minus fitbestfit (119895) minus fit (119894)

gt rand or fit (119895) gt fit (119894)

0 otherwise(12)

After production of the new position of the CPs if anycomponent of the solution vector swerves off the allowablebounds correct its position using the harmony search-basedhandling approach as described in [23]

The charged memory (CM) is used to save a number ofthe best solutions up to the iterationThe better new solutionsare included in the CM and the worst ones are excluded fromthe CM The flowchart of the CSS algorithm is illustrated inFigure 1

4 Objective Function

The CSS algorithm attempts to minimize an objective assess-ment function for the best solution to a given problem Thisfunction is used to provide a measure of how individualshave performed in the problem domain In the case of aminimization problem the fittest individuals will have thelowest numerical value of the associated objective functionIn this study the statement for the objective function is givenas

119865 = 119891 (119889) (13)

where 119889 = 1198891 1198892 119889119873 are damage parameters at the 119873elements

To create the objective function 119865 it is essential touse some kind of output variables of the structure thatare sufficiently sensitive to the damage parameter beingidentified in order to avoid ill conditioning problems Themode shapes and the natural frequencies can be obtained by

Incomplete noisy dataComplete noisy dataActual damage

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

Element number

Dam

age (

)

Figure 5 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 2

9 128 11 117 10

20

20

1221

21

19

19

17

17

15

15

13

13

18

18

16

16

14

14

22 9

75

5

3 3

1

1

10

86

6

44

22

25

24

23

160

4060

Figure 6 A 25-bar Bowstring truss

modal analysis methods Natural frequencies are relativelyeasy to measure and have been used by many researchers

To identify localized damage because of the greaterexperience variations in the locality of the affected areamodeshapes offer a better option (Salane and Baldwin 1990 [24]Salawu and Williams 1995 [25] Ndambi et al 2002 [26]) Itis necessary to note that the success of this process depends onthe quality and place selection (for test) of the measurementswhich are able to reflect the damage

Because of this assuming that only a few natural fre-quencies and mode shapes of the lower modes are availablefrequency objective functions from incomplete data areconsidered in this paper to detect damage

5 Numerical Simulation Study

In this section two examples consisting of a ten-element can-tilever beam a Bowstring plane truss and a three-story three-bay unbraced frame structures are presented to examine thecharged system search algorithm The final results are thencompared to the solutions of particle swarm optimization

6 Shock and Vibration

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

04

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 7 The convergence history of the Bowstring truss for the CSS and PSOPC algorithms

Incomplete noisy dataComplete noisy dataActual damage

0 2 4 6 8 10 12 14 16 18 20 22 240

5

10

15

20

25

30

Element number

Dam

age (

)

Figure 8 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 1

with passive congregation algorithm to demonstrate theperformance of this work

In CSS algorithm the effect of the pervious velocity andthe resultant force affecting a CP can decrease or increasebased on the values of the 119896V and 119896119886 defined as [22]

119896V = 119888(1 minusiter

itermax)

119896119886 = 119888(1 +iter

itermax)

(14)

where iter is the iteration number itermaxis the maximumnumber of the iterations and 119888 is set to 1

For the CSS algorithm a population of 16 CPs and forPSOPC algorithm a population of 50 particles are used for all

Incomplete noisy dataComplete noisy dataActual damage

Element number0 2 4 6 8 10 12 14 16 18 20 22 24

0

5

10

15

20

25

30

35

40

45D

amag

e (

)

Figure 9 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 2

Table 1 Simulated damage scenarios in cantilever beam

Element number Scenario 1 Scenario 21 1035 206 107 359

the examples the stop criterion is considered as maximumnumber of 3000 and 5000 analyses in first and two otherexamples respectively

Shock and Vibration 7

1

2

13

23 24

14 16 181715

22 25 26 27

19 2120

28 3029

32 3331 34 35 36 37 3938

3 6 9 12

5 8 11

4 7 104m

35m

8m 7m 9m

35m

W12 times 65

W14 times 132

Figure 10 Geometry of unbraced plane frame

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

01

02

03

04

05

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Number of analyses

0

005

01

015

02

025

03

035

04

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 11 The convergence history of the 39-element frame for the CSS and PSOPC algorithms

The algorithm has been implemented in the commercialMATLAB software and because of the stochastic nature of thealgorithms each example is independently solved ten times

Damage values have been limited to a region between 0and 1

Example 1 A cantilever steel beam is shown in Figure 2 Forthe reason of modal analyzing the beam was divided into 10two-dimensional beam elements with 20 degrees of freedom

The section of the beam is W12 times 65 with mechanicalproperties of

119860 = 00123 m2 (191 in2)

119868 = 2218 times 10minus4 m4 (533 in4)

119864 = 207 times 109Nm2

120588 = 7860 kgm3

The measured modal responses of the beam before and afterdamage were created using the proposed forward solver(for each scenario) Instead of experimental measurementsnumerically generated measurements were used to estimatethe proposed inverse procedure In this example the first tennatural frequencies are used for damage detection Two dif-ferent simulated damage scenarios are considered (Table 1)

These damage scenarios are considered to represent theeffect of severity of damage (stiffness reduction) numberof damaged elements and contribution of damage elementson the results The results of damage detecting of CSS andPSOPC algorithms are given in Table 2 In order to evaluatethe performance of the methods used the summation ofestimated error of damage detection was calculated (Table 2)that is presented as follows

Error = sum(119889119894119860 minus 119889119894119878) (15)

8 Shock and Vibration

Table 2 Results of damage detection of the 10-element beam using CSS and PSOPC algorithms

Elementnumber

Scenario 1 Scenario 2

CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0000 0 0000 0101 01 0105

2 0000 0 0000 0003 0 0000

3 0000 0 0000 0001 0 0000

4 0000 0 0000 0000 0 0000

5 0000 0 0000 0185 02 0169

6 0100 01 0100 0016 0 0037

7 0000 0 0000 0349 035 0348

8 0000 0 0000 0002 0 0000

9 0000 0 0000 0001 0 0000

10 0000 0 0000 0000 0 0000

Error 00002 00000 00385 00752

Table 3 The values of objective function of the 10-element beam for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 34 times 10

minus0652 times 10

minus0628 times 10

minus0627 times 10

minus0683 times 10

minus0665 times 10

minus0641 times 10

minus0638 times 10

minus0619 times 10

minus0644 times 10

minus06 100PSOPC 25 times 10

minus0713 times 10

minus0799 times 10

minus0814 times 10

minus0721 times 10

minus0746 times 10

minus0726 times 10

minus0716 times 10

minus0688 times 10

minus0811 times 10

minus07 100Scenario 2

CSS 00008 00001 00009 00004 00006 00002 00005 00008 00009 00006 80PSOPC 00003 00013 00056 00004 00074 00074 00080 00011 00006 00085 40

Table 4 Cross-sectional areas of truss elements

Member Area (cm2)1ndash6 187ndash12 1513ndash17 1018ndash25 12

Table 5 Simulated damage scenarios in Bowstring truss

Element number Scenario 1 Scenario 235 108 251011 4024 20

where 119889119894119860 and 119889119894119878 are the actual and estimated damage of the

119894th element using the presented method respectivelyIn Table 3 the values of the objective function for CSS

and PSOPC algorithms in independent 10 runs are given theconvergence history for the 10-element beam in scenarios 1and 2 is shown in Figure 3

It is seen in Table 2 that in the first scenario where thenumber of damaged elements is low the damage identifi-cation is well by both algorithms in the second scenariothe algorithm PSOPC has some wrong in damage detectionso that there is difference between the obtained damage viathis algorithm and the actual intensity of damage in thefifth element and the undamaged sixth element mistakenlyhas been detected to be damaged Meanwhile the locationand amount of damage in structure are obtained by the CSSalgorithm with acceptable accuracy In order to investigatethe noise effects on the results of the CSSmethod 015 noisein measurement is considered for the natural frequencies

Diagrams of the CSS algorithmrsquos damage detection withcomplete and incomplete dynamic noisy data are plotted inFigures 4 and 5

As the results show this method is able to detect thelocation andmagnitude of damaged elements in all scenariosusing complete and incomplete noisy data

Example 2 A Bowstring plane truss with 25 elements isshown in Figure 6 The properties of members of this struc-ture are 119864 = 207 times 0

9Nm2 and 120588 = 7860 kgm3 and cross-sectional areas of elements are given in Table 4

Three scenarios for this truss are considered according toTable 5

Shock and Vibration 9

Table 6 Results of damage detection of the 25-bar Bowstring truss using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0009 0 0000 0000 0 00002 0000 0 0000 0000 0 00003 0000 0 0000 0000 0 00004 0000 0 0000 0000 0 00005 0000 0 0000 0098 01 00956 0000 0 0000 0000 0 00007 0000 0 0000 0000 0 00008 0246 025 0250 0000 0 00009 0001 0 0000 0000 0 000010 0000 0 0000 0000 0 000011 0000 0 0000 0402 04 042612 0000 0 0000 0000 0 000013 0000 0 0000 0000 0 000014 0001 0 0000 0000 0 000015 0000 0 0000 0000 0 000016 0000 0 0000 0000 0 000017 0000 0 0000 0000 0 000018 0000 0 0000 0005 0 000019 0000 0 0000 0004 0 007520 0000 0 0000 0001 0 000021 0000 0 0000 0000 0 000022 0000 0 0000 0000 0 000023 0000 0 0000 0000 0 000024 0000 0 0000 0185 02 000025 0000 0 0000 0014 0 0265Error 00085 00000 00119 01609

In Table 6 the results of two algorithms in all scenarios arecompared Only the first ten natural frequencies are used fordamage detection

The comparison of the objective function for CSS andPSOPC algorithms is made in Table 7 and the convergencehistory for the Bowstring truss in scenarios 1 and 2 is shownin Figure 7

In Figures 8 and 9 the results of CSS algorithm in damagedetection of this truss that is affected by noise are shown

Table 7 and Figures 7 8 and 9 show better results indamage detection by CSS algorithm rather than PSOPCalgorithm and also by applying noise in structures

Example 3 A three-story three-bay frame as shown inFigure 10 is used to verify the damage detection methodexplained in this paper The number of elements and nodesis 39 and 34 respectively

For unbraced plane frame problem all columns areW14 times 132 (119860 = 0025m2 (388 in2) and 119868 = 6386 times

10minus4m4 (1530 in4)) and all beams are W12 times 65 Youngrsquos

modulus is 119864 = 207GPa (30 000 ksi) Poissonrsquos ratio is ] =

03 and the mass density is 120588 = 7780 kgm3 (0000728 lb minuss2in4)

In this example two scenarios from the aspect of elementnumber and its level of damage were assumed that are givenin Table 8 Between ninety natural frequencies only the firsttwenty natural frequencies are utilized for damage detectionin the first scenario

The severity of damage detection by PSOPC and CSSalgorithms in different members of the frame and in eachscenario is shown separately in Table 9 In Table 10 theobtained objective function for both algorithms in each runis given also the convergence history is shown in Figure 11

Also the results of the CSS algorithm damage detectionwith noisy data are plotted in the diagrams of Figures 12 and13

The results of damage detection with two algorithmsin this large frame show that in case of only one memberdamaged there is more accuracy with PSOPC algorithm inidentifying of desired member and obtaining the amount ofdamage than the CSS algorithm It is observed that increasingthe number of injured elements in the structures reducesthe result precision with algorithms From results errors itcan be concluded that the PSOPC algorithm has problemin detecting the location of some of the damaged structuralelements so that in the third scenario the tenth and twenty-fourth members which are injured are diagnosed as healthy

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

many methods that have been introduced to correctly findout the place and severity of structural damage

Perera and Torres 2006 studied a nondestructive globaldamage detection and assessment methodology based onthe changes in frequencies and mode shapes of vibrationof a structural system It was shown that the proposed GAyields a suitable damage places and severity detection fromtraditional methods [8]

A two-stage method for determining place and severityof multiple structural damage by combining the adaptiveneurofuzzy inference system (ANFIS) and particle swarmoptimization (PSO) has been proposed by Fallahian andSeyedpoor [9] In a studyKoh andDyke [10] have determinedthe place and severity of multiple damage by iterativelysearching for a combination of structural responses thatmaximizes a correlation coefficient named the multipledamage location assurance criterion (MDLAC) via a geneticalgorithm (GA) Damage detection by a hybrid techniqueincluding a real-parameter genetic algorithm and grey rela-tion analysis has been presented by He and Hwang [11]They used first a grey relation analysis to exclude impossibledamage places such that the number of design variables couldbe reduced Second a real-parameter genetic algorithm wascombined with simulated annealing for finding the actualdamage The damage identification of a beam-like structurehas been formulated as an optimization problem and a GAhas been employed to find the damage place and severityby minimizing the cost function which is based on thedifference of measured and calculated natural frequencies Atwo-stage method for determining the place and severity ofmultiple structural damage by using an information fusiontechnique and a GA has been presented by Guo and Li [12]In the first stage the damage is localized by using evidencetheory and then a microsearch genetic algorithm (MSGA)has been planned to determine the damage size Yu and Chendetermined the place and severity of damage in a bridgeFirst two objective functions are defined One is defined asminimizing the sum of differences between the modal databefore and after damage in traditional wayThe other is newlydefined based on modal flexibility which is combined withanother function able to predict damage location Secondlyan improved particle swarm optimization (PSO) algorithmis developed based on the macroeconomic strategies andused to solve the multiple-objective optimization problemon bridge damage identification The results show that theprocedure is very promising for locating and quantifyingdamaged elements of bridge structures and considerablyimproves predictions based on the modal flexibility as wellas the PSO method [13] An approach for detecting damagein structural members based on continuum damage modelusing an algorithmnamedBig Bang-Big Crunch (BB-BC) haspresented suitable accuracy in results [14]

In recent years the use of metaheuristic methods hasincreased and great efforts to increase the power of algo-rithms (Efficiency) and reduce optimization time (moreconvergence speed) have been done Recently a new androbust algorithm was presented by Kaveh and Talatahari socalled charged system search [15] This algorithm is based onsome laws from electrostatics and the Newtonian mechanics

The charged system search (CSS) utilizes a number of chargedparticles (CP) which affect each other based on their fitnessvalues and separation distances consideringCoulombGaussandNewtonian lawsThe resultant forces and themotion lawsdetermine the new location of the particles The harmonysearch-based handling approach is utilized for controlling thevariable constraint

In this paper the CSS algorithm is utilized to predict thedamage place and severity for different types of structuresThe structure is modeled with the finite element method andthe damage identification is performed at element level withincomplete modal data

In order to show the power of the CSS algorithm in struc-tural damage detection the obtained results are comparedto the results of the PSOPC algorithm [16] PSOPC algo-rithm is a particle swarm optimization method with passivecongregationrsquos capacity [17] High abilities of this algorithmwith acceptable results in structural damage detection areillustrated [16]

The results show that the proposed method is capableof detecting the location and severity of diagnosis evenin large-scale structures with a large number of damagedelements it achieved acceptable resultsThe effect of the noiseis considered by assigning the noise in natural frequencieson the results of the CSS algorithm The graphs indicate thatthe noise in input data may reduce the accuracy of damagedetection

2 Theory

Existing Structural damage detection techniques can beclassified into two major categories the dynamic and staticmethods [18] Both methods are based on the finite elementutilizing experimental test data requiring dynamic and statictest data correspondingly In addition dynamic methodshave shown their advantages in comparison with static onesThe natural frequencies of a structure can be considered tobe valuable with the dynamic data Determining the levelof correlation between the calculated and predicted naturalfrequencies can provide a simple tool for finding the place andseverity of structural damage [19] Structural damage detec-tion is calculated with changes in structural characteristicsand it is possible with global evaluating Doebling et al havediscussed vibration-based techniques in a literature review[20] In this section the construction of dynamics of damagedstructures is discussed

The parameter vector used for evaluating the correlationcoefficients (the ratio of the changes first 119899119891 natural fre-quency Δ119865 due to structural damage) is

Δ119865 =

119865119873 minus 119865119864

119865119873

(1)

where 119865119873 and 119865119864 denote the natural frequency vectors of theundamaged and damaged structure Similarly the parametervector predicted from an analytical model can be definedcorrespondingly as

120575119865 (119883) =

119865119873 minus 119865 (119883)

119865119873

(2)

Shock and Vibration 3

where 119865(119883) is a natural frequency vector that can be pre-dicted from an analytic model and 119883 = [119909

1 1199092 119909

119899]

represents a damage variable vector containing the damageseverity of all 119899 structural elements Given a pair of param-eter vectors one can estimate the level of correlation inseveral ways An efficient way is to evaluate a correlation-based index termed themultiples damage location assurancecriterion (MDLAC) and covered in the following form [19]

MDLAC (119883) =10038161003816100381610038161003816Δ119865119879sdot 120575119865 (119883)

10038161003816100381610038161003816

2

(Δ119865119879sdot Δ119865) (120575119865

119879(119883) sdot 120575119865 (119883))

0 lt MDLAC lt 1

(3)

Two frequency change vectors are compared with MDLACone calculated from the structural tests and the other froma structural model analysis When the vector of analyticalfrequencies becomes the same as the frequency vector ofthe damaged structure MDLAC will be maximal That is119865(119883) = 119865 so considering this theory can be used to finda set of damage variables maximizing the MDLAC using anoptimization algorithm

Find 119883 = [119909119894 1199092 119909119899]

Maximize 119908 (119883) = MDLAC (119883) (4)

The damage severity can take values only from the set thatis given from [ 0 1 ] a set of continuous values Moreoverthe objective function that should be maximized is 119908 Asmentioned the damage occurrence in a structural elementdecreases the element stiffness Thus one of the methods forthe damage identification problem is simulation damage bydecreasing one of the stiffness parameters of the element suchas the modulus of elasticity (119864) cross-sectional area (119860) andinertia moment (119868) In this study the damage variables aredefined via a relative reduction of the elasticity modulus ofan element as

119870119889119894 = (1 minus 119909119894)119870

ℎ119894

(5)

119870119889119894 is the stiffness matrix of damaged element 119894 and119870ℎ119894 is the

stiffness matrix of healthy element 119894119864 is the primary modulus of elasticity and 119864119894 is the final

modulus of elasticity of the 119894th element The MDLAC as anobjective function for the optimization algorithm is moresensitive to damaged elements than undamaged elementsIt means that this method can find the true place of thedamaged elements but it may find an undamaged element asa damaged one Therefore in this study a new function andnew optimization algorithm is discussed the new function ispresented as [21]

obj (119883) = 1

119899119891

119899119891

sum

119894=1

min (119891119909119894 119891119864119894)max (119891119909119894 119891119864119894)

(6)

where119891119909119894119891119864119894 are the 119894th components of vectors 119865(119883) and 119865119864correspondingly

The obj(119883) function can rapidly find the locations ofhealthy elements when compared to theMDLAC however it

is very probable that it finds a damaged element as a healthyone Therefore in this study a combinational function of(3) and (6) called here the efficient correlation-based index(ECBI) is used as [21]

ECBI (119883) = 1

2

(MDLAC (119883) + obj (119883)) (7)

3 Charged System Search Algorithm

A new type of metaheuristic algorithms is introduced byKaveh and Talatahari [15] which is called charge systemsearch The charged system search (CSS) is based onCoulomb and Gauss laws from electrical physics and thegoverning laws of motion from the Newtonian mechanicsIn this algorithm each agent is a charged particle (CP) EachCP is considered as a charged sphere which exerts an electricforce on other CPs according to Coulomb and Gauss lawsThe resultant forces and the motion laws determine the newlocation of the CPs [22] The new positions of the chargedparticles in the first iteration are determined randomly andfor next iterations are obtained as follows

119883119895new = rand1198951 sdot 119896119886 sdot119865119895

119898119895

sdot Δ1199052

+ rand1198952 sdot 119896V sdot 119881119895old sdot Δ119905 + 119883119895old

119881119895new =119883119895new minus 119883119895old

Δ119905

(8)

where 119870119886 and 119870V are the acceleration and the velocitycoefficients respectively rand1198951 and rand1198952 are two randomuniformly distributed in the range (0 1) and the resultantforces vector for 119895th CP 119865119895 is calculated as

119865119895 = 119902119894 sum

119894119894 = 119895

(

119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119901119894119895 (119883119894 minus 119883119895)

119895 = 1 2 119873

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(9)

where the magnitude of charge for each CP 119902119894 is defined as

119902119894 =fit (119894) minus fitworstfitbest minus fitworst

119894 = 1 2 119873 (10)

where fitbest and fitworst are the best and the worst fitness ofall the CPs fit(119894) is the fitness of the agent 119894 and119873 is the totalnumber of charged particlesThe separation distance betweentwo CPs 119903119894119895 is obtained as follows

119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

10038171003817100381710038171003817

10038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576

(11)

where119883119894 and119883119895 are the positions of the 119894th and 119895th CPs119883bestis the position of the best current CP and 120576 is a small positive

4 Shock and Vibration

Search

No

Yes

No

Yes

Terminationcriterion

Step 1 Step 2

Step 3

Step 4

Step 1 Step 2

Step 3

Initialize specification of problem and algorithm

parameters and determinethe initial charged particles

Find the values of the objective function for the

CPs and rank of CPs in an increasing order

Store some of the best CPs in

the charged memory (CM)

Determine the probability of

moving and force vector for each CP

Determine the new positions and

velocities of the CPs

Correct the CP positions using an algorithm based on HS if the CP does not satisfy

the side limits

Evaluate the objective function and rank the

CPs according to their quality

Include the better vectors in the CM and exclude the worst ones

from the CM

Stop

Are the new CPs better than the stored ones

in CM

Initialization

Figure 1 Flowchart of the CSS algorithm

1 2 96 8 1043 5 7

1007mW12 times 65

Figure 2 A cantilever beam geometry

0 500 1000 1500 2000 2500 30000

005

01

015

02

025

Number of analyses

Obj

ectiv

e fun

ctio

n

CSSPSOPC

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 30000

005

01

015

02

025

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 3 The convergence history of the 10-element beam for the CSS and PSOPC algorithms

Shock and Vibration 5

0 1 2 3 4 5 6 7 8 9 100

5

10

15

Element number

Dam

age (

)

Incomplete noisy dataComplete noisy dataActual damage

Figure 4 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 1

numberThe probability of moving each CP toward the otherCPs is determined using the following function

119901119894119895 =

1

fit (119894) minus fitbestfit (119895) minus fit (119894)

gt rand or fit (119895) gt fit (119894)

0 otherwise(12)

After production of the new position of the CPs if anycomponent of the solution vector swerves off the allowablebounds correct its position using the harmony search-basedhandling approach as described in [23]

The charged memory (CM) is used to save a number ofthe best solutions up to the iterationThe better new solutionsare included in the CM and the worst ones are excluded fromthe CM The flowchart of the CSS algorithm is illustrated inFigure 1

4 Objective Function

The CSS algorithm attempts to minimize an objective assess-ment function for the best solution to a given problem Thisfunction is used to provide a measure of how individualshave performed in the problem domain In the case of aminimization problem the fittest individuals will have thelowest numerical value of the associated objective functionIn this study the statement for the objective function is givenas

119865 = 119891 (119889) (13)

where 119889 = 1198891 1198892 119889119873 are damage parameters at the 119873elements

To create the objective function 119865 it is essential touse some kind of output variables of the structure thatare sufficiently sensitive to the damage parameter beingidentified in order to avoid ill conditioning problems Themode shapes and the natural frequencies can be obtained by

Incomplete noisy dataComplete noisy dataActual damage

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

Element number

Dam

age (

)

Figure 5 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 2

9 128 11 117 10

20

20

1221

21

19

19

17

17

15

15

13

13

18

18

16

16

14

14

22 9

75

5

3 3

1

1

10

86

6

44

22

25

24

23

160

4060

Figure 6 A 25-bar Bowstring truss

modal analysis methods Natural frequencies are relativelyeasy to measure and have been used by many researchers

To identify localized damage because of the greaterexperience variations in the locality of the affected areamodeshapes offer a better option (Salane and Baldwin 1990 [24]Salawu and Williams 1995 [25] Ndambi et al 2002 [26]) Itis necessary to note that the success of this process depends onthe quality and place selection (for test) of the measurementswhich are able to reflect the damage

Because of this assuming that only a few natural fre-quencies and mode shapes of the lower modes are availablefrequency objective functions from incomplete data areconsidered in this paper to detect damage

5 Numerical Simulation Study

In this section two examples consisting of a ten-element can-tilever beam a Bowstring plane truss and a three-story three-bay unbraced frame structures are presented to examine thecharged system search algorithm The final results are thencompared to the solutions of particle swarm optimization

6 Shock and Vibration

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

04

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 7 The convergence history of the Bowstring truss for the CSS and PSOPC algorithms

Incomplete noisy dataComplete noisy dataActual damage

0 2 4 6 8 10 12 14 16 18 20 22 240

5

10

15

20

25

30

Element number

Dam

age (

)

Figure 8 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 1

with passive congregation algorithm to demonstrate theperformance of this work

In CSS algorithm the effect of the pervious velocity andthe resultant force affecting a CP can decrease or increasebased on the values of the 119896V and 119896119886 defined as [22]

119896V = 119888(1 minusiter

itermax)

119896119886 = 119888(1 +iter

itermax)

(14)

where iter is the iteration number itermaxis the maximumnumber of the iterations and 119888 is set to 1

For the CSS algorithm a population of 16 CPs and forPSOPC algorithm a population of 50 particles are used for all

Incomplete noisy dataComplete noisy dataActual damage

Element number0 2 4 6 8 10 12 14 16 18 20 22 24

0

5

10

15

20

25

30

35

40

45D

amag

e (

)

Figure 9 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 2

Table 1 Simulated damage scenarios in cantilever beam

Element number Scenario 1 Scenario 21 1035 206 107 359

the examples the stop criterion is considered as maximumnumber of 3000 and 5000 analyses in first and two otherexamples respectively

Shock and Vibration 7

1

2

13

23 24

14 16 181715

22 25 26 27

19 2120

28 3029

32 3331 34 35 36 37 3938

3 6 9 12

5 8 11

4 7 104m

35m

8m 7m 9m

35m

W12 times 65

W14 times 132

Figure 10 Geometry of unbraced plane frame

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

01

02

03

04

05

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Number of analyses

0

005

01

015

02

025

03

035

04

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 11 The convergence history of the 39-element frame for the CSS and PSOPC algorithms

The algorithm has been implemented in the commercialMATLAB software and because of the stochastic nature of thealgorithms each example is independently solved ten times

Damage values have been limited to a region between 0and 1

Example 1 A cantilever steel beam is shown in Figure 2 Forthe reason of modal analyzing the beam was divided into 10two-dimensional beam elements with 20 degrees of freedom

The section of the beam is W12 times 65 with mechanicalproperties of

119860 = 00123 m2 (191 in2)

119868 = 2218 times 10minus4 m4 (533 in4)

119864 = 207 times 109Nm2

120588 = 7860 kgm3

The measured modal responses of the beam before and afterdamage were created using the proposed forward solver(for each scenario) Instead of experimental measurementsnumerically generated measurements were used to estimatethe proposed inverse procedure In this example the first tennatural frequencies are used for damage detection Two dif-ferent simulated damage scenarios are considered (Table 1)

These damage scenarios are considered to represent theeffect of severity of damage (stiffness reduction) numberof damaged elements and contribution of damage elementson the results The results of damage detecting of CSS andPSOPC algorithms are given in Table 2 In order to evaluatethe performance of the methods used the summation ofestimated error of damage detection was calculated (Table 2)that is presented as follows

Error = sum(119889119894119860 minus 119889119894119878) (15)

8 Shock and Vibration

Table 2 Results of damage detection of the 10-element beam using CSS and PSOPC algorithms

Elementnumber

Scenario 1 Scenario 2

CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0000 0 0000 0101 01 0105

2 0000 0 0000 0003 0 0000

3 0000 0 0000 0001 0 0000

4 0000 0 0000 0000 0 0000

5 0000 0 0000 0185 02 0169

6 0100 01 0100 0016 0 0037

7 0000 0 0000 0349 035 0348

8 0000 0 0000 0002 0 0000

9 0000 0 0000 0001 0 0000

10 0000 0 0000 0000 0 0000

Error 00002 00000 00385 00752

Table 3 The values of objective function of the 10-element beam for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 34 times 10

minus0652 times 10

minus0628 times 10

minus0627 times 10

minus0683 times 10

minus0665 times 10

minus0641 times 10

minus0638 times 10

minus0619 times 10

minus0644 times 10

minus06 100PSOPC 25 times 10

minus0713 times 10

minus0799 times 10

minus0814 times 10

minus0721 times 10

minus0746 times 10

minus0726 times 10

minus0716 times 10

minus0688 times 10

minus0811 times 10

minus07 100Scenario 2

CSS 00008 00001 00009 00004 00006 00002 00005 00008 00009 00006 80PSOPC 00003 00013 00056 00004 00074 00074 00080 00011 00006 00085 40

Table 4 Cross-sectional areas of truss elements

Member Area (cm2)1ndash6 187ndash12 1513ndash17 1018ndash25 12

Table 5 Simulated damage scenarios in Bowstring truss

Element number Scenario 1 Scenario 235 108 251011 4024 20

where 119889119894119860 and 119889119894119878 are the actual and estimated damage of the

119894th element using the presented method respectivelyIn Table 3 the values of the objective function for CSS

and PSOPC algorithms in independent 10 runs are given theconvergence history for the 10-element beam in scenarios 1and 2 is shown in Figure 3

It is seen in Table 2 that in the first scenario where thenumber of damaged elements is low the damage identifi-cation is well by both algorithms in the second scenariothe algorithm PSOPC has some wrong in damage detectionso that there is difference between the obtained damage viathis algorithm and the actual intensity of damage in thefifth element and the undamaged sixth element mistakenlyhas been detected to be damaged Meanwhile the locationand amount of damage in structure are obtained by the CSSalgorithm with acceptable accuracy In order to investigatethe noise effects on the results of the CSSmethod 015 noisein measurement is considered for the natural frequencies

Diagrams of the CSS algorithmrsquos damage detection withcomplete and incomplete dynamic noisy data are plotted inFigures 4 and 5

As the results show this method is able to detect thelocation andmagnitude of damaged elements in all scenariosusing complete and incomplete noisy data

Example 2 A Bowstring plane truss with 25 elements isshown in Figure 6 The properties of members of this struc-ture are 119864 = 207 times 0

9Nm2 and 120588 = 7860 kgm3 and cross-sectional areas of elements are given in Table 4

Three scenarios for this truss are considered according toTable 5

Shock and Vibration 9

Table 6 Results of damage detection of the 25-bar Bowstring truss using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0009 0 0000 0000 0 00002 0000 0 0000 0000 0 00003 0000 0 0000 0000 0 00004 0000 0 0000 0000 0 00005 0000 0 0000 0098 01 00956 0000 0 0000 0000 0 00007 0000 0 0000 0000 0 00008 0246 025 0250 0000 0 00009 0001 0 0000 0000 0 000010 0000 0 0000 0000 0 000011 0000 0 0000 0402 04 042612 0000 0 0000 0000 0 000013 0000 0 0000 0000 0 000014 0001 0 0000 0000 0 000015 0000 0 0000 0000 0 000016 0000 0 0000 0000 0 000017 0000 0 0000 0000 0 000018 0000 0 0000 0005 0 000019 0000 0 0000 0004 0 007520 0000 0 0000 0001 0 000021 0000 0 0000 0000 0 000022 0000 0 0000 0000 0 000023 0000 0 0000 0000 0 000024 0000 0 0000 0185 02 000025 0000 0 0000 0014 0 0265Error 00085 00000 00119 01609

In Table 6 the results of two algorithms in all scenarios arecompared Only the first ten natural frequencies are used fordamage detection

The comparison of the objective function for CSS andPSOPC algorithms is made in Table 7 and the convergencehistory for the Bowstring truss in scenarios 1 and 2 is shownin Figure 7

In Figures 8 and 9 the results of CSS algorithm in damagedetection of this truss that is affected by noise are shown

Table 7 and Figures 7 8 and 9 show better results indamage detection by CSS algorithm rather than PSOPCalgorithm and also by applying noise in structures

Example 3 A three-story three-bay frame as shown inFigure 10 is used to verify the damage detection methodexplained in this paper The number of elements and nodesis 39 and 34 respectively

For unbraced plane frame problem all columns areW14 times 132 (119860 = 0025m2 (388 in2) and 119868 = 6386 times

10minus4m4 (1530 in4)) and all beams are W12 times 65 Youngrsquos

modulus is 119864 = 207GPa (30 000 ksi) Poissonrsquos ratio is ] =

03 and the mass density is 120588 = 7780 kgm3 (0000728 lb minuss2in4)

In this example two scenarios from the aspect of elementnumber and its level of damage were assumed that are givenin Table 8 Between ninety natural frequencies only the firsttwenty natural frequencies are utilized for damage detectionin the first scenario

The severity of damage detection by PSOPC and CSSalgorithms in different members of the frame and in eachscenario is shown separately in Table 9 In Table 10 theobtained objective function for both algorithms in each runis given also the convergence history is shown in Figure 11

Also the results of the CSS algorithm damage detectionwith noisy data are plotted in the diagrams of Figures 12 and13

The results of damage detection with two algorithmsin this large frame show that in case of only one memberdamaged there is more accuracy with PSOPC algorithm inidentifying of desired member and obtaining the amount ofdamage than the CSS algorithm It is observed that increasingthe number of injured elements in the structures reducesthe result precision with algorithms From results errors itcan be concluded that the PSOPC algorithm has problemin detecting the location of some of the damaged structuralelements so that in the third scenario the tenth and twenty-fourth members which are injured are diagnosed as healthy

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

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RotatingMachinery

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VLSI Design

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Shock and Vibration

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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International Journal of

Shock and Vibration 3

where 119865(119883) is a natural frequency vector that can be pre-dicted from an analytic model and 119883 = [119909

1 1199092 119909

119899]

represents a damage variable vector containing the damageseverity of all 119899 structural elements Given a pair of param-eter vectors one can estimate the level of correlation inseveral ways An efficient way is to evaluate a correlation-based index termed themultiples damage location assurancecriterion (MDLAC) and covered in the following form [19]

MDLAC (119883) =10038161003816100381610038161003816Δ119865119879sdot 120575119865 (119883)

10038161003816100381610038161003816

2

(Δ119865119879sdot Δ119865) (120575119865

119879(119883) sdot 120575119865 (119883))

0 lt MDLAC lt 1

(3)

Two frequency change vectors are compared with MDLACone calculated from the structural tests and the other froma structural model analysis When the vector of analyticalfrequencies becomes the same as the frequency vector ofthe damaged structure MDLAC will be maximal That is119865(119883) = 119865 so considering this theory can be used to finda set of damage variables maximizing the MDLAC using anoptimization algorithm

Find 119883 = [119909119894 1199092 119909119899]

Maximize 119908 (119883) = MDLAC (119883) (4)

The damage severity can take values only from the set thatis given from [ 0 1 ] a set of continuous values Moreoverthe objective function that should be maximized is 119908 Asmentioned the damage occurrence in a structural elementdecreases the element stiffness Thus one of the methods forthe damage identification problem is simulation damage bydecreasing one of the stiffness parameters of the element suchas the modulus of elasticity (119864) cross-sectional area (119860) andinertia moment (119868) In this study the damage variables aredefined via a relative reduction of the elasticity modulus ofan element as

119870119889119894 = (1 minus 119909119894)119870

ℎ119894

(5)

119870119889119894 is the stiffness matrix of damaged element 119894 and119870ℎ119894 is the

stiffness matrix of healthy element 119894119864 is the primary modulus of elasticity and 119864119894 is the final

modulus of elasticity of the 119894th element The MDLAC as anobjective function for the optimization algorithm is moresensitive to damaged elements than undamaged elementsIt means that this method can find the true place of thedamaged elements but it may find an undamaged element asa damaged one Therefore in this study a new function andnew optimization algorithm is discussed the new function ispresented as [21]

obj (119883) = 1

119899119891

119899119891

sum

119894=1

min (119891119909119894 119891119864119894)max (119891119909119894 119891119864119894)

(6)

where119891119909119894119891119864119894 are the 119894th components of vectors 119865(119883) and 119865119864correspondingly

The obj(119883) function can rapidly find the locations ofhealthy elements when compared to theMDLAC however it

is very probable that it finds a damaged element as a healthyone Therefore in this study a combinational function of(3) and (6) called here the efficient correlation-based index(ECBI) is used as [21]

ECBI (119883) = 1

2

(MDLAC (119883) + obj (119883)) (7)

3 Charged System Search Algorithm

A new type of metaheuristic algorithms is introduced byKaveh and Talatahari [15] which is called charge systemsearch The charged system search (CSS) is based onCoulomb and Gauss laws from electrical physics and thegoverning laws of motion from the Newtonian mechanicsIn this algorithm each agent is a charged particle (CP) EachCP is considered as a charged sphere which exerts an electricforce on other CPs according to Coulomb and Gauss lawsThe resultant forces and the motion laws determine the newlocation of the CPs [22] The new positions of the chargedparticles in the first iteration are determined randomly andfor next iterations are obtained as follows

119883119895new = rand1198951 sdot 119896119886 sdot119865119895

119898119895

sdot Δ1199052

+ rand1198952 sdot 119896V sdot 119881119895old sdot Δ119905 + 119883119895old

119881119895new =119883119895new minus 119883119895old

Δ119905

(8)

where 119870119886 and 119870V are the acceleration and the velocitycoefficients respectively rand1198951 and rand1198952 are two randomuniformly distributed in the range (0 1) and the resultantforces vector for 119895th CP 119865119895 is calculated as

119865119895 = 119902119894 sum

119894119894 = 119895

(

119902119894

1198863119903119894119895 sdot 1198941 +

119902119894

1199032119894119895

sdot 1198942)119901119894119895 (119883119894 minus 119883119895)

119895 = 1 2 119873

1198941 = 1 1198942 = 0 lArrrArr 119903119894119895 lt 119886

1198941 = 0 1198942 = 1 lArrrArr 119903119894119895 ge 119886

(9)

where the magnitude of charge for each CP 119902119894 is defined as

119902119894 =fit (119894) minus fitworstfitbest minus fitworst

119894 = 1 2 119873 (10)

where fitbest and fitworst are the best and the worst fitness ofall the CPs fit(119894) is the fitness of the agent 119894 and119873 is the totalnumber of charged particlesThe separation distance betweentwo CPs 119903119894119895 is obtained as follows

119903119894119895 =

10038171003817100381710038171003817119883119894 minus 119883119895

10038171003817100381710038171003817

10038171003817100381710038171003817(119883119894 + 119883119895) 2 minus 119883best

10038171003817100381710038171003817+ 120576

(11)

where119883119894 and119883119895 are the positions of the 119894th and 119895th CPs119883bestis the position of the best current CP and 120576 is a small positive

4 Shock and Vibration

Search

No

Yes

No

Yes

Terminationcriterion

Step 1 Step 2

Step 3

Step 4

Step 1 Step 2

Step 3

Initialize specification of problem and algorithm

parameters and determinethe initial charged particles

Find the values of the objective function for the

CPs and rank of CPs in an increasing order

Store some of the best CPs in

the charged memory (CM)

Determine the probability of

moving and force vector for each CP

Determine the new positions and

velocities of the CPs

Correct the CP positions using an algorithm based on HS if the CP does not satisfy

the side limits

Evaluate the objective function and rank the

CPs according to their quality

Include the better vectors in the CM and exclude the worst ones

from the CM

Stop

Are the new CPs better than the stored ones

in CM

Initialization

Figure 1 Flowchart of the CSS algorithm

1 2 96 8 1043 5 7

1007mW12 times 65

Figure 2 A cantilever beam geometry

0 500 1000 1500 2000 2500 30000

005

01

015

02

025

Number of analyses

Obj

ectiv

e fun

ctio

n

CSSPSOPC

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 30000

005

01

015

02

025

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 3 The convergence history of the 10-element beam for the CSS and PSOPC algorithms

Shock and Vibration 5

0 1 2 3 4 5 6 7 8 9 100

5

10

15

Element number

Dam

age (

)

Incomplete noisy dataComplete noisy dataActual damage

Figure 4 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 1

numberThe probability of moving each CP toward the otherCPs is determined using the following function

119901119894119895 =

1

fit (119894) minus fitbestfit (119895) minus fit (119894)

gt rand or fit (119895) gt fit (119894)

0 otherwise(12)

After production of the new position of the CPs if anycomponent of the solution vector swerves off the allowablebounds correct its position using the harmony search-basedhandling approach as described in [23]

The charged memory (CM) is used to save a number ofthe best solutions up to the iterationThe better new solutionsare included in the CM and the worst ones are excluded fromthe CM The flowchart of the CSS algorithm is illustrated inFigure 1

4 Objective Function

The CSS algorithm attempts to minimize an objective assess-ment function for the best solution to a given problem Thisfunction is used to provide a measure of how individualshave performed in the problem domain In the case of aminimization problem the fittest individuals will have thelowest numerical value of the associated objective functionIn this study the statement for the objective function is givenas

119865 = 119891 (119889) (13)

where 119889 = 1198891 1198892 119889119873 are damage parameters at the 119873elements

To create the objective function 119865 it is essential touse some kind of output variables of the structure thatare sufficiently sensitive to the damage parameter beingidentified in order to avoid ill conditioning problems Themode shapes and the natural frequencies can be obtained by

Incomplete noisy dataComplete noisy dataActual damage

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

Element number

Dam

age (

)

Figure 5 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 2

9 128 11 117 10

20

20

1221

21

19

19

17

17

15

15

13

13

18

18

16

16

14

14

22 9

75

5

3 3

1

1

10

86

6

44

22

25

24

23

160

4060

Figure 6 A 25-bar Bowstring truss

modal analysis methods Natural frequencies are relativelyeasy to measure and have been used by many researchers

To identify localized damage because of the greaterexperience variations in the locality of the affected areamodeshapes offer a better option (Salane and Baldwin 1990 [24]Salawu and Williams 1995 [25] Ndambi et al 2002 [26]) Itis necessary to note that the success of this process depends onthe quality and place selection (for test) of the measurementswhich are able to reflect the damage

Because of this assuming that only a few natural fre-quencies and mode shapes of the lower modes are availablefrequency objective functions from incomplete data areconsidered in this paper to detect damage

5 Numerical Simulation Study

In this section two examples consisting of a ten-element can-tilever beam a Bowstring plane truss and a three-story three-bay unbraced frame structures are presented to examine thecharged system search algorithm The final results are thencompared to the solutions of particle swarm optimization

6 Shock and Vibration

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

04

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 7 The convergence history of the Bowstring truss for the CSS and PSOPC algorithms

Incomplete noisy dataComplete noisy dataActual damage

0 2 4 6 8 10 12 14 16 18 20 22 240

5

10

15

20

25

30

Element number

Dam

age (

)

Figure 8 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 1

with passive congregation algorithm to demonstrate theperformance of this work

In CSS algorithm the effect of the pervious velocity andthe resultant force affecting a CP can decrease or increasebased on the values of the 119896V and 119896119886 defined as [22]

119896V = 119888(1 minusiter

itermax)

119896119886 = 119888(1 +iter

itermax)

(14)

where iter is the iteration number itermaxis the maximumnumber of the iterations and 119888 is set to 1

For the CSS algorithm a population of 16 CPs and forPSOPC algorithm a population of 50 particles are used for all

Incomplete noisy dataComplete noisy dataActual damage

Element number0 2 4 6 8 10 12 14 16 18 20 22 24

0

5

10

15

20

25

30

35

40

45D

amag

e (

)

Figure 9 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 2

Table 1 Simulated damage scenarios in cantilever beam

Element number Scenario 1 Scenario 21 1035 206 107 359

the examples the stop criterion is considered as maximumnumber of 3000 and 5000 analyses in first and two otherexamples respectively

Shock and Vibration 7

1

2

13

23 24

14 16 181715

22 25 26 27

19 2120

28 3029

32 3331 34 35 36 37 3938

3 6 9 12

5 8 11

4 7 104m

35m

8m 7m 9m

35m

W12 times 65

W14 times 132

Figure 10 Geometry of unbraced plane frame

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

01

02

03

04

05

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Number of analyses

0

005

01

015

02

025

03

035

04

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 11 The convergence history of the 39-element frame for the CSS and PSOPC algorithms

The algorithm has been implemented in the commercialMATLAB software and because of the stochastic nature of thealgorithms each example is independently solved ten times

Damage values have been limited to a region between 0and 1

Example 1 A cantilever steel beam is shown in Figure 2 Forthe reason of modal analyzing the beam was divided into 10two-dimensional beam elements with 20 degrees of freedom

The section of the beam is W12 times 65 with mechanicalproperties of

119860 = 00123 m2 (191 in2)

119868 = 2218 times 10minus4 m4 (533 in4)

119864 = 207 times 109Nm2

120588 = 7860 kgm3

The measured modal responses of the beam before and afterdamage were created using the proposed forward solver(for each scenario) Instead of experimental measurementsnumerically generated measurements were used to estimatethe proposed inverse procedure In this example the first tennatural frequencies are used for damage detection Two dif-ferent simulated damage scenarios are considered (Table 1)

These damage scenarios are considered to represent theeffect of severity of damage (stiffness reduction) numberof damaged elements and contribution of damage elementson the results The results of damage detecting of CSS andPSOPC algorithms are given in Table 2 In order to evaluatethe performance of the methods used the summation ofestimated error of damage detection was calculated (Table 2)that is presented as follows

Error = sum(119889119894119860 minus 119889119894119878) (15)

8 Shock and Vibration

Table 2 Results of damage detection of the 10-element beam using CSS and PSOPC algorithms

Elementnumber

Scenario 1 Scenario 2

CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0000 0 0000 0101 01 0105

2 0000 0 0000 0003 0 0000

3 0000 0 0000 0001 0 0000

4 0000 0 0000 0000 0 0000

5 0000 0 0000 0185 02 0169

6 0100 01 0100 0016 0 0037

7 0000 0 0000 0349 035 0348

8 0000 0 0000 0002 0 0000

9 0000 0 0000 0001 0 0000

10 0000 0 0000 0000 0 0000

Error 00002 00000 00385 00752

Table 3 The values of objective function of the 10-element beam for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 34 times 10

minus0652 times 10

minus0628 times 10

minus0627 times 10

minus0683 times 10

minus0665 times 10

minus0641 times 10

minus0638 times 10

minus0619 times 10

minus0644 times 10

minus06 100PSOPC 25 times 10

minus0713 times 10

minus0799 times 10

minus0814 times 10

minus0721 times 10

minus0746 times 10

minus0726 times 10

minus0716 times 10

minus0688 times 10

minus0811 times 10

minus07 100Scenario 2

CSS 00008 00001 00009 00004 00006 00002 00005 00008 00009 00006 80PSOPC 00003 00013 00056 00004 00074 00074 00080 00011 00006 00085 40

Table 4 Cross-sectional areas of truss elements

Member Area (cm2)1ndash6 187ndash12 1513ndash17 1018ndash25 12

Table 5 Simulated damage scenarios in Bowstring truss

Element number Scenario 1 Scenario 235 108 251011 4024 20

where 119889119894119860 and 119889119894119878 are the actual and estimated damage of the

119894th element using the presented method respectivelyIn Table 3 the values of the objective function for CSS

and PSOPC algorithms in independent 10 runs are given theconvergence history for the 10-element beam in scenarios 1and 2 is shown in Figure 3

It is seen in Table 2 that in the first scenario where thenumber of damaged elements is low the damage identifi-cation is well by both algorithms in the second scenariothe algorithm PSOPC has some wrong in damage detectionso that there is difference between the obtained damage viathis algorithm and the actual intensity of damage in thefifth element and the undamaged sixth element mistakenlyhas been detected to be damaged Meanwhile the locationand amount of damage in structure are obtained by the CSSalgorithm with acceptable accuracy In order to investigatethe noise effects on the results of the CSSmethod 015 noisein measurement is considered for the natural frequencies

Diagrams of the CSS algorithmrsquos damage detection withcomplete and incomplete dynamic noisy data are plotted inFigures 4 and 5

As the results show this method is able to detect thelocation andmagnitude of damaged elements in all scenariosusing complete and incomplete noisy data

Example 2 A Bowstring plane truss with 25 elements isshown in Figure 6 The properties of members of this struc-ture are 119864 = 207 times 0

9Nm2 and 120588 = 7860 kgm3 and cross-sectional areas of elements are given in Table 4

Three scenarios for this truss are considered according toTable 5

Shock and Vibration 9

Table 6 Results of damage detection of the 25-bar Bowstring truss using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0009 0 0000 0000 0 00002 0000 0 0000 0000 0 00003 0000 0 0000 0000 0 00004 0000 0 0000 0000 0 00005 0000 0 0000 0098 01 00956 0000 0 0000 0000 0 00007 0000 0 0000 0000 0 00008 0246 025 0250 0000 0 00009 0001 0 0000 0000 0 000010 0000 0 0000 0000 0 000011 0000 0 0000 0402 04 042612 0000 0 0000 0000 0 000013 0000 0 0000 0000 0 000014 0001 0 0000 0000 0 000015 0000 0 0000 0000 0 000016 0000 0 0000 0000 0 000017 0000 0 0000 0000 0 000018 0000 0 0000 0005 0 000019 0000 0 0000 0004 0 007520 0000 0 0000 0001 0 000021 0000 0 0000 0000 0 000022 0000 0 0000 0000 0 000023 0000 0 0000 0000 0 000024 0000 0 0000 0185 02 000025 0000 0 0000 0014 0 0265Error 00085 00000 00119 01609

In Table 6 the results of two algorithms in all scenarios arecompared Only the first ten natural frequencies are used fordamage detection

The comparison of the objective function for CSS andPSOPC algorithms is made in Table 7 and the convergencehistory for the Bowstring truss in scenarios 1 and 2 is shownin Figure 7

In Figures 8 and 9 the results of CSS algorithm in damagedetection of this truss that is affected by noise are shown

Table 7 and Figures 7 8 and 9 show better results indamage detection by CSS algorithm rather than PSOPCalgorithm and also by applying noise in structures

Example 3 A three-story three-bay frame as shown inFigure 10 is used to verify the damage detection methodexplained in this paper The number of elements and nodesis 39 and 34 respectively

For unbraced plane frame problem all columns areW14 times 132 (119860 = 0025m2 (388 in2) and 119868 = 6386 times

10minus4m4 (1530 in4)) and all beams are W12 times 65 Youngrsquos

modulus is 119864 = 207GPa (30 000 ksi) Poissonrsquos ratio is ] =

03 and the mass density is 120588 = 7780 kgm3 (0000728 lb minuss2in4)

In this example two scenarios from the aspect of elementnumber and its level of damage were assumed that are givenin Table 8 Between ninety natural frequencies only the firsttwenty natural frequencies are utilized for damage detectionin the first scenario

The severity of damage detection by PSOPC and CSSalgorithms in different members of the frame and in eachscenario is shown separately in Table 9 In Table 10 theobtained objective function for both algorithms in each runis given also the convergence history is shown in Figure 11

Also the results of the CSS algorithm damage detectionwith noisy data are plotted in the diagrams of Figures 12 and13

The results of damage detection with two algorithmsin this large frame show that in case of only one memberdamaged there is more accuracy with PSOPC algorithm inidentifying of desired member and obtaining the amount ofdamage than the CSS algorithm It is observed that increasingthe number of injured elements in the structures reducesthe result precision with algorithms From results errors itcan be concluded that the PSOPC algorithm has problemin detecting the location of some of the damaged structuralelements so that in the third scenario the tenth and twenty-fourth members which are injured are diagnosed as healthy

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

Search

No

Yes

No

Yes

Terminationcriterion

Step 1 Step 2

Step 3

Step 4

Step 1 Step 2

Step 3

Initialize specification of problem and algorithm

parameters and determinethe initial charged particles

Find the values of the objective function for the

CPs and rank of CPs in an increasing order

Store some of the best CPs in

the charged memory (CM)

Determine the probability of

moving and force vector for each CP

Determine the new positions and

velocities of the CPs

Correct the CP positions using an algorithm based on HS if the CP does not satisfy

the side limits

Evaluate the objective function and rank the

CPs according to their quality

Include the better vectors in the CM and exclude the worst ones

from the CM

Stop

Are the new CPs better than the stored ones

in CM

Initialization

Figure 1 Flowchart of the CSS algorithm

1 2 96 8 1043 5 7

1007mW12 times 65

Figure 2 A cantilever beam geometry

0 500 1000 1500 2000 2500 30000

005

01

015

02

025

Number of analyses

Obj

ectiv

e fun

ctio

n

CSSPSOPC

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 30000

005

01

015

02

025

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 3 The convergence history of the 10-element beam for the CSS and PSOPC algorithms

Shock and Vibration 5

0 1 2 3 4 5 6 7 8 9 100

5

10

15

Element number

Dam

age (

)

Incomplete noisy dataComplete noisy dataActual damage

Figure 4 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 1

numberThe probability of moving each CP toward the otherCPs is determined using the following function

119901119894119895 =

1

fit (119894) minus fitbestfit (119895) minus fit (119894)

gt rand or fit (119895) gt fit (119894)

0 otherwise(12)

After production of the new position of the CPs if anycomponent of the solution vector swerves off the allowablebounds correct its position using the harmony search-basedhandling approach as described in [23]

The charged memory (CM) is used to save a number ofthe best solutions up to the iterationThe better new solutionsare included in the CM and the worst ones are excluded fromthe CM The flowchart of the CSS algorithm is illustrated inFigure 1

4 Objective Function

The CSS algorithm attempts to minimize an objective assess-ment function for the best solution to a given problem Thisfunction is used to provide a measure of how individualshave performed in the problem domain In the case of aminimization problem the fittest individuals will have thelowest numerical value of the associated objective functionIn this study the statement for the objective function is givenas

119865 = 119891 (119889) (13)

where 119889 = 1198891 1198892 119889119873 are damage parameters at the 119873elements

To create the objective function 119865 it is essential touse some kind of output variables of the structure thatare sufficiently sensitive to the damage parameter beingidentified in order to avoid ill conditioning problems Themode shapes and the natural frequencies can be obtained by

Incomplete noisy dataComplete noisy dataActual damage

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

Element number

Dam

age (

)

Figure 5 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 2

9 128 11 117 10

20

20

1221

21

19

19

17

17

15

15

13

13

18

18

16

16

14

14

22 9

75

5

3 3

1

1

10

86

6

44

22

25

24

23

160

4060

Figure 6 A 25-bar Bowstring truss

modal analysis methods Natural frequencies are relativelyeasy to measure and have been used by many researchers

To identify localized damage because of the greaterexperience variations in the locality of the affected areamodeshapes offer a better option (Salane and Baldwin 1990 [24]Salawu and Williams 1995 [25] Ndambi et al 2002 [26]) Itis necessary to note that the success of this process depends onthe quality and place selection (for test) of the measurementswhich are able to reflect the damage

Because of this assuming that only a few natural fre-quencies and mode shapes of the lower modes are availablefrequency objective functions from incomplete data areconsidered in this paper to detect damage

5 Numerical Simulation Study

In this section two examples consisting of a ten-element can-tilever beam a Bowstring plane truss and a three-story three-bay unbraced frame structures are presented to examine thecharged system search algorithm The final results are thencompared to the solutions of particle swarm optimization

6 Shock and Vibration

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

04

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 7 The convergence history of the Bowstring truss for the CSS and PSOPC algorithms

Incomplete noisy dataComplete noisy dataActual damage

0 2 4 6 8 10 12 14 16 18 20 22 240

5

10

15

20

25

30

Element number

Dam

age (

)

Figure 8 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 1

with passive congregation algorithm to demonstrate theperformance of this work

In CSS algorithm the effect of the pervious velocity andthe resultant force affecting a CP can decrease or increasebased on the values of the 119896V and 119896119886 defined as [22]

119896V = 119888(1 minusiter

itermax)

119896119886 = 119888(1 +iter

itermax)

(14)

where iter is the iteration number itermaxis the maximumnumber of the iterations and 119888 is set to 1

For the CSS algorithm a population of 16 CPs and forPSOPC algorithm a population of 50 particles are used for all

Incomplete noisy dataComplete noisy dataActual damage

Element number0 2 4 6 8 10 12 14 16 18 20 22 24

0

5

10

15

20

25

30

35

40

45D

amag

e (

)

Figure 9 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 2

Table 1 Simulated damage scenarios in cantilever beam

Element number Scenario 1 Scenario 21 1035 206 107 359

the examples the stop criterion is considered as maximumnumber of 3000 and 5000 analyses in first and two otherexamples respectively

Shock and Vibration 7

1

2

13

23 24

14 16 181715

22 25 26 27

19 2120

28 3029

32 3331 34 35 36 37 3938

3 6 9 12

5 8 11

4 7 104m

35m

8m 7m 9m

35m

W12 times 65

W14 times 132

Figure 10 Geometry of unbraced plane frame

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

01

02

03

04

05

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Number of analyses

0

005

01

015

02

025

03

035

04

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 11 The convergence history of the 39-element frame for the CSS and PSOPC algorithms

The algorithm has been implemented in the commercialMATLAB software and because of the stochastic nature of thealgorithms each example is independently solved ten times

Damage values have been limited to a region between 0and 1

Example 1 A cantilever steel beam is shown in Figure 2 Forthe reason of modal analyzing the beam was divided into 10two-dimensional beam elements with 20 degrees of freedom

The section of the beam is W12 times 65 with mechanicalproperties of

119860 = 00123 m2 (191 in2)

119868 = 2218 times 10minus4 m4 (533 in4)

119864 = 207 times 109Nm2

120588 = 7860 kgm3

The measured modal responses of the beam before and afterdamage were created using the proposed forward solver(for each scenario) Instead of experimental measurementsnumerically generated measurements were used to estimatethe proposed inverse procedure In this example the first tennatural frequencies are used for damage detection Two dif-ferent simulated damage scenarios are considered (Table 1)

These damage scenarios are considered to represent theeffect of severity of damage (stiffness reduction) numberof damaged elements and contribution of damage elementson the results The results of damage detecting of CSS andPSOPC algorithms are given in Table 2 In order to evaluatethe performance of the methods used the summation ofestimated error of damage detection was calculated (Table 2)that is presented as follows

Error = sum(119889119894119860 minus 119889119894119878) (15)

8 Shock and Vibration

Table 2 Results of damage detection of the 10-element beam using CSS and PSOPC algorithms

Elementnumber

Scenario 1 Scenario 2

CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0000 0 0000 0101 01 0105

2 0000 0 0000 0003 0 0000

3 0000 0 0000 0001 0 0000

4 0000 0 0000 0000 0 0000

5 0000 0 0000 0185 02 0169

6 0100 01 0100 0016 0 0037

7 0000 0 0000 0349 035 0348

8 0000 0 0000 0002 0 0000

9 0000 0 0000 0001 0 0000

10 0000 0 0000 0000 0 0000

Error 00002 00000 00385 00752

Table 3 The values of objective function of the 10-element beam for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 34 times 10

minus0652 times 10

minus0628 times 10

minus0627 times 10

minus0683 times 10

minus0665 times 10

minus0641 times 10

minus0638 times 10

minus0619 times 10

minus0644 times 10

minus06 100PSOPC 25 times 10

minus0713 times 10

minus0799 times 10

minus0814 times 10

minus0721 times 10

minus0746 times 10

minus0726 times 10

minus0716 times 10

minus0688 times 10

minus0811 times 10

minus07 100Scenario 2

CSS 00008 00001 00009 00004 00006 00002 00005 00008 00009 00006 80PSOPC 00003 00013 00056 00004 00074 00074 00080 00011 00006 00085 40

Table 4 Cross-sectional areas of truss elements

Member Area (cm2)1ndash6 187ndash12 1513ndash17 1018ndash25 12

Table 5 Simulated damage scenarios in Bowstring truss

Element number Scenario 1 Scenario 235 108 251011 4024 20

where 119889119894119860 and 119889119894119878 are the actual and estimated damage of the

119894th element using the presented method respectivelyIn Table 3 the values of the objective function for CSS

and PSOPC algorithms in independent 10 runs are given theconvergence history for the 10-element beam in scenarios 1and 2 is shown in Figure 3

It is seen in Table 2 that in the first scenario where thenumber of damaged elements is low the damage identifi-cation is well by both algorithms in the second scenariothe algorithm PSOPC has some wrong in damage detectionso that there is difference between the obtained damage viathis algorithm and the actual intensity of damage in thefifth element and the undamaged sixth element mistakenlyhas been detected to be damaged Meanwhile the locationand amount of damage in structure are obtained by the CSSalgorithm with acceptable accuracy In order to investigatethe noise effects on the results of the CSSmethod 015 noisein measurement is considered for the natural frequencies

Diagrams of the CSS algorithmrsquos damage detection withcomplete and incomplete dynamic noisy data are plotted inFigures 4 and 5

As the results show this method is able to detect thelocation andmagnitude of damaged elements in all scenariosusing complete and incomplete noisy data

Example 2 A Bowstring plane truss with 25 elements isshown in Figure 6 The properties of members of this struc-ture are 119864 = 207 times 0

9Nm2 and 120588 = 7860 kgm3 and cross-sectional areas of elements are given in Table 4

Three scenarios for this truss are considered according toTable 5

Shock and Vibration 9

Table 6 Results of damage detection of the 25-bar Bowstring truss using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0009 0 0000 0000 0 00002 0000 0 0000 0000 0 00003 0000 0 0000 0000 0 00004 0000 0 0000 0000 0 00005 0000 0 0000 0098 01 00956 0000 0 0000 0000 0 00007 0000 0 0000 0000 0 00008 0246 025 0250 0000 0 00009 0001 0 0000 0000 0 000010 0000 0 0000 0000 0 000011 0000 0 0000 0402 04 042612 0000 0 0000 0000 0 000013 0000 0 0000 0000 0 000014 0001 0 0000 0000 0 000015 0000 0 0000 0000 0 000016 0000 0 0000 0000 0 000017 0000 0 0000 0000 0 000018 0000 0 0000 0005 0 000019 0000 0 0000 0004 0 007520 0000 0 0000 0001 0 000021 0000 0 0000 0000 0 000022 0000 0 0000 0000 0 000023 0000 0 0000 0000 0 000024 0000 0 0000 0185 02 000025 0000 0 0000 0014 0 0265Error 00085 00000 00119 01609

In Table 6 the results of two algorithms in all scenarios arecompared Only the first ten natural frequencies are used fordamage detection

The comparison of the objective function for CSS andPSOPC algorithms is made in Table 7 and the convergencehistory for the Bowstring truss in scenarios 1 and 2 is shownin Figure 7

In Figures 8 and 9 the results of CSS algorithm in damagedetection of this truss that is affected by noise are shown

Table 7 and Figures 7 8 and 9 show better results indamage detection by CSS algorithm rather than PSOPCalgorithm and also by applying noise in structures

Example 3 A three-story three-bay frame as shown inFigure 10 is used to verify the damage detection methodexplained in this paper The number of elements and nodesis 39 and 34 respectively

For unbraced plane frame problem all columns areW14 times 132 (119860 = 0025m2 (388 in2) and 119868 = 6386 times

10minus4m4 (1530 in4)) and all beams are W12 times 65 Youngrsquos

modulus is 119864 = 207GPa (30 000 ksi) Poissonrsquos ratio is ] =

03 and the mass density is 120588 = 7780 kgm3 (0000728 lb minuss2in4)

In this example two scenarios from the aspect of elementnumber and its level of damage were assumed that are givenin Table 8 Between ninety natural frequencies only the firsttwenty natural frequencies are utilized for damage detectionin the first scenario

The severity of damage detection by PSOPC and CSSalgorithms in different members of the frame and in eachscenario is shown separately in Table 9 In Table 10 theobtained objective function for both algorithms in each runis given also the convergence history is shown in Figure 11

Also the results of the CSS algorithm damage detectionwith noisy data are plotted in the diagrams of Figures 12 and13

The results of damage detection with two algorithmsin this large frame show that in case of only one memberdamaged there is more accuracy with PSOPC algorithm inidentifying of desired member and obtaining the amount ofdamage than the CSS algorithm It is observed that increasingthe number of injured elements in the structures reducesthe result precision with algorithms From results errors itcan be concluded that the PSOPC algorithm has problemin detecting the location of some of the damaged structuralelements so that in the third scenario the tenth and twenty-fourth members which are injured are diagnosed as healthy

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

0 1 2 3 4 5 6 7 8 9 100

5

10

15

Element number

Dam

age (

)

Incomplete noisy dataComplete noisy dataActual damage

Figure 4 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 1

numberThe probability of moving each CP toward the otherCPs is determined using the following function

119901119894119895 =

1

fit (119894) minus fitbestfit (119895) minus fit (119894)

gt rand or fit (119895) gt fit (119894)

0 otherwise(12)

After production of the new position of the CPs if anycomponent of the solution vector swerves off the allowablebounds correct its position using the harmony search-basedhandling approach as described in [23]

The charged memory (CM) is used to save a number ofthe best solutions up to the iterationThe better new solutionsare included in the CM and the worst ones are excluded fromthe CM The flowchart of the CSS algorithm is illustrated inFigure 1

4 Objective Function

The CSS algorithm attempts to minimize an objective assess-ment function for the best solution to a given problem Thisfunction is used to provide a measure of how individualshave performed in the problem domain In the case of aminimization problem the fittest individuals will have thelowest numerical value of the associated objective functionIn this study the statement for the objective function is givenas

119865 = 119891 (119889) (13)

where 119889 = 1198891 1198892 119889119873 are damage parameters at the 119873elements

To create the objective function 119865 it is essential touse some kind of output variables of the structure thatare sufficiently sensitive to the damage parameter beingidentified in order to avoid ill conditioning problems Themode shapes and the natural frequencies can be obtained by

Incomplete noisy dataComplete noisy dataActual damage

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

Element number

Dam

age (

)

Figure 5 Damage distribution of cantilever beam using the com-plete and incomplete noisy data in Scenario 2

9 128 11 117 10

20

20

1221

21

19

19

17

17

15

15

13

13

18

18

16

16

14

14

22 9

75

5

3 3

1

1

10

86

6

44

22

25

24

23

160

4060

Figure 6 A 25-bar Bowstring truss

modal analysis methods Natural frequencies are relativelyeasy to measure and have been used by many researchers

To identify localized damage because of the greaterexperience variations in the locality of the affected areamodeshapes offer a better option (Salane and Baldwin 1990 [24]Salawu and Williams 1995 [25] Ndambi et al 2002 [26]) Itis necessary to note that the success of this process depends onthe quality and place selection (for test) of the measurementswhich are able to reflect the damage

Because of this assuming that only a few natural fre-quencies and mode shapes of the lower modes are availablefrequency objective functions from incomplete data areconsidered in this paper to detect damage

5 Numerical Simulation Study

In this section two examples consisting of a ten-element can-tilever beam a Bowstring plane truss and a three-story three-bay unbraced frame structures are presented to examine thecharged system search algorithm The final results are thencompared to the solutions of particle swarm optimization

6 Shock and Vibration

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

04

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 7 The convergence history of the Bowstring truss for the CSS and PSOPC algorithms

Incomplete noisy dataComplete noisy dataActual damage

0 2 4 6 8 10 12 14 16 18 20 22 240

5

10

15

20

25

30

Element number

Dam

age (

)

Figure 8 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 1

with passive congregation algorithm to demonstrate theperformance of this work

In CSS algorithm the effect of the pervious velocity andthe resultant force affecting a CP can decrease or increasebased on the values of the 119896V and 119896119886 defined as [22]

119896V = 119888(1 minusiter

itermax)

119896119886 = 119888(1 +iter

itermax)

(14)

where iter is the iteration number itermaxis the maximumnumber of the iterations and 119888 is set to 1

For the CSS algorithm a population of 16 CPs and forPSOPC algorithm a population of 50 particles are used for all

Incomplete noisy dataComplete noisy dataActual damage

Element number0 2 4 6 8 10 12 14 16 18 20 22 24

0

5

10

15

20

25

30

35

40

45D

amag

e (

)

Figure 9 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 2

Table 1 Simulated damage scenarios in cantilever beam

Element number Scenario 1 Scenario 21 1035 206 107 359

the examples the stop criterion is considered as maximumnumber of 3000 and 5000 analyses in first and two otherexamples respectively

Shock and Vibration 7

1

2

13

23 24

14 16 181715

22 25 26 27

19 2120

28 3029

32 3331 34 35 36 37 3938

3 6 9 12

5 8 11

4 7 104m

35m

8m 7m 9m

35m

W12 times 65

W14 times 132

Figure 10 Geometry of unbraced plane frame

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

01

02

03

04

05

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Number of analyses

0

005

01

015

02

025

03

035

04

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 11 The convergence history of the 39-element frame for the CSS and PSOPC algorithms

The algorithm has been implemented in the commercialMATLAB software and because of the stochastic nature of thealgorithms each example is independently solved ten times

Damage values have been limited to a region between 0and 1

Example 1 A cantilever steel beam is shown in Figure 2 Forthe reason of modal analyzing the beam was divided into 10two-dimensional beam elements with 20 degrees of freedom

The section of the beam is W12 times 65 with mechanicalproperties of

119860 = 00123 m2 (191 in2)

119868 = 2218 times 10minus4 m4 (533 in4)

119864 = 207 times 109Nm2

120588 = 7860 kgm3

The measured modal responses of the beam before and afterdamage were created using the proposed forward solver(for each scenario) Instead of experimental measurementsnumerically generated measurements were used to estimatethe proposed inverse procedure In this example the first tennatural frequencies are used for damage detection Two dif-ferent simulated damage scenarios are considered (Table 1)

These damage scenarios are considered to represent theeffect of severity of damage (stiffness reduction) numberof damaged elements and contribution of damage elementson the results The results of damage detecting of CSS andPSOPC algorithms are given in Table 2 In order to evaluatethe performance of the methods used the summation ofestimated error of damage detection was calculated (Table 2)that is presented as follows

Error = sum(119889119894119860 minus 119889119894119878) (15)

8 Shock and Vibration

Table 2 Results of damage detection of the 10-element beam using CSS and PSOPC algorithms

Elementnumber

Scenario 1 Scenario 2

CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0000 0 0000 0101 01 0105

2 0000 0 0000 0003 0 0000

3 0000 0 0000 0001 0 0000

4 0000 0 0000 0000 0 0000

5 0000 0 0000 0185 02 0169

6 0100 01 0100 0016 0 0037

7 0000 0 0000 0349 035 0348

8 0000 0 0000 0002 0 0000

9 0000 0 0000 0001 0 0000

10 0000 0 0000 0000 0 0000

Error 00002 00000 00385 00752

Table 3 The values of objective function of the 10-element beam for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 34 times 10

minus0652 times 10

minus0628 times 10

minus0627 times 10

minus0683 times 10

minus0665 times 10

minus0641 times 10

minus0638 times 10

minus0619 times 10

minus0644 times 10

minus06 100PSOPC 25 times 10

minus0713 times 10

minus0799 times 10

minus0814 times 10

minus0721 times 10

minus0746 times 10

minus0726 times 10

minus0716 times 10

minus0688 times 10

minus0811 times 10

minus07 100Scenario 2

CSS 00008 00001 00009 00004 00006 00002 00005 00008 00009 00006 80PSOPC 00003 00013 00056 00004 00074 00074 00080 00011 00006 00085 40

Table 4 Cross-sectional areas of truss elements

Member Area (cm2)1ndash6 187ndash12 1513ndash17 1018ndash25 12

Table 5 Simulated damage scenarios in Bowstring truss

Element number Scenario 1 Scenario 235 108 251011 4024 20

where 119889119894119860 and 119889119894119878 are the actual and estimated damage of the

119894th element using the presented method respectivelyIn Table 3 the values of the objective function for CSS

and PSOPC algorithms in independent 10 runs are given theconvergence history for the 10-element beam in scenarios 1and 2 is shown in Figure 3

It is seen in Table 2 that in the first scenario where thenumber of damaged elements is low the damage identifi-cation is well by both algorithms in the second scenariothe algorithm PSOPC has some wrong in damage detectionso that there is difference between the obtained damage viathis algorithm and the actual intensity of damage in thefifth element and the undamaged sixth element mistakenlyhas been detected to be damaged Meanwhile the locationand amount of damage in structure are obtained by the CSSalgorithm with acceptable accuracy In order to investigatethe noise effects on the results of the CSSmethod 015 noisein measurement is considered for the natural frequencies

Diagrams of the CSS algorithmrsquos damage detection withcomplete and incomplete dynamic noisy data are plotted inFigures 4 and 5

As the results show this method is able to detect thelocation andmagnitude of damaged elements in all scenariosusing complete and incomplete noisy data

Example 2 A Bowstring plane truss with 25 elements isshown in Figure 6 The properties of members of this struc-ture are 119864 = 207 times 0

9Nm2 and 120588 = 7860 kgm3 and cross-sectional areas of elements are given in Table 4

Three scenarios for this truss are considered according toTable 5

Shock and Vibration 9

Table 6 Results of damage detection of the 25-bar Bowstring truss using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0009 0 0000 0000 0 00002 0000 0 0000 0000 0 00003 0000 0 0000 0000 0 00004 0000 0 0000 0000 0 00005 0000 0 0000 0098 01 00956 0000 0 0000 0000 0 00007 0000 0 0000 0000 0 00008 0246 025 0250 0000 0 00009 0001 0 0000 0000 0 000010 0000 0 0000 0000 0 000011 0000 0 0000 0402 04 042612 0000 0 0000 0000 0 000013 0000 0 0000 0000 0 000014 0001 0 0000 0000 0 000015 0000 0 0000 0000 0 000016 0000 0 0000 0000 0 000017 0000 0 0000 0000 0 000018 0000 0 0000 0005 0 000019 0000 0 0000 0004 0 007520 0000 0 0000 0001 0 000021 0000 0 0000 0000 0 000022 0000 0 0000 0000 0 000023 0000 0 0000 0000 0 000024 0000 0 0000 0185 02 000025 0000 0 0000 0014 0 0265Error 00085 00000 00119 01609

In Table 6 the results of two algorithms in all scenarios arecompared Only the first ten natural frequencies are used fordamage detection

The comparison of the objective function for CSS andPSOPC algorithms is made in Table 7 and the convergencehistory for the Bowstring truss in scenarios 1 and 2 is shownin Figure 7

In Figures 8 and 9 the results of CSS algorithm in damagedetection of this truss that is affected by noise are shown

Table 7 and Figures 7 8 and 9 show better results indamage detection by CSS algorithm rather than PSOPCalgorithm and also by applying noise in structures

Example 3 A three-story three-bay frame as shown inFigure 10 is used to verify the damage detection methodexplained in this paper The number of elements and nodesis 39 and 34 respectively

For unbraced plane frame problem all columns areW14 times 132 (119860 = 0025m2 (388 in2) and 119868 = 6386 times

10minus4m4 (1530 in4)) and all beams are W12 times 65 Youngrsquos

modulus is 119864 = 207GPa (30 000 ksi) Poissonrsquos ratio is ] =

03 and the mass density is 120588 = 7780 kgm3 (0000728 lb minuss2in4)

In this example two scenarios from the aspect of elementnumber and its level of damage were assumed that are givenin Table 8 Between ninety natural frequencies only the firsttwenty natural frequencies are utilized for damage detectionin the first scenario

The severity of damage detection by PSOPC and CSSalgorithms in different members of the frame and in eachscenario is shown separately in Table 9 In Table 10 theobtained objective function for both algorithms in each runis given also the convergence history is shown in Figure 11

Also the results of the CSS algorithm damage detectionwith noisy data are plotted in the diagrams of Figures 12 and13

The results of damage detection with two algorithmsin this large frame show that in case of only one memberdamaged there is more accuracy with PSOPC algorithm inidentifying of desired member and obtaining the amount ofdamage than the CSS algorithm It is observed that increasingthe number of injured elements in the structures reducesthe result precision with algorithms From results errors itcan be concluded that the PSOPC algorithm has problemin detecting the location of some of the damaged structuralelements so that in the third scenario the tenth and twenty-fourth members which are injured are diagnosed as healthy

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

6 Shock and Vibration

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

04

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

005

01

015

02

025

03

035

Number of analyses

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 7 The convergence history of the Bowstring truss for the CSS and PSOPC algorithms

Incomplete noisy dataComplete noisy dataActual damage

0 2 4 6 8 10 12 14 16 18 20 22 240

5

10

15

20

25

30

Element number

Dam

age (

)

Figure 8 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 1

with passive congregation algorithm to demonstrate theperformance of this work

In CSS algorithm the effect of the pervious velocity andthe resultant force affecting a CP can decrease or increasebased on the values of the 119896V and 119896119886 defined as [22]

119896V = 119888(1 minusiter

itermax)

119896119886 = 119888(1 +iter

itermax)

(14)

where iter is the iteration number itermaxis the maximumnumber of the iterations and 119888 is set to 1

For the CSS algorithm a population of 16 CPs and forPSOPC algorithm a population of 50 particles are used for all

Incomplete noisy dataComplete noisy dataActual damage

Element number0 2 4 6 8 10 12 14 16 18 20 22 24

0

5

10

15

20

25

30

35

40

45D

amag

e (

)

Figure 9 Damage distribution of Bowstring truss using the com-plete and incomplete noisy data in Scenario 2

Table 1 Simulated damage scenarios in cantilever beam

Element number Scenario 1 Scenario 21 1035 206 107 359

the examples the stop criterion is considered as maximumnumber of 3000 and 5000 analyses in first and two otherexamples respectively

Shock and Vibration 7

1

2

13

23 24

14 16 181715

22 25 26 27

19 2120

28 3029

32 3331 34 35 36 37 3938

3 6 9 12

5 8 11

4 7 104m

35m

8m 7m 9m

35m

W12 times 65

W14 times 132

Figure 10 Geometry of unbraced plane frame

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

01

02

03

04

05

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Number of analyses

0

005

01

015

02

025

03

035

04

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 11 The convergence history of the 39-element frame for the CSS and PSOPC algorithms

The algorithm has been implemented in the commercialMATLAB software and because of the stochastic nature of thealgorithms each example is independently solved ten times

Damage values have been limited to a region between 0and 1

Example 1 A cantilever steel beam is shown in Figure 2 Forthe reason of modal analyzing the beam was divided into 10two-dimensional beam elements with 20 degrees of freedom

The section of the beam is W12 times 65 with mechanicalproperties of

119860 = 00123 m2 (191 in2)

119868 = 2218 times 10minus4 m4 (533 in4)

119864 = 207 times 109Nm2

120588 = 7860 kgm3

The measured modal responses of the beam before and afterdamage were created using the proposed forward solver(for each scenario) Instead of experimental measurementsnumerically generated measurements were used to estimatethe proposed inverse procedure In this example the first tennatural frequencies are used for damage detection Two dif-ferent simulated damage scenarios are considered (Table 1)

These damage scenarios are considered to represent theeffect of severity of damage (stiffness reduction) numberof damaged elements and contribution of damage elementson the results The results of damage detecting of CSS andPSOPC algorithms are given in Table 2 In order to evaluatethe performance of the methods used the summation ofestimated error of damage detection was calculated (Table 2)that is presented as follows

Error = sum(119889119894119860 minus 119889119894119878) (15)

8 Shock and Vibration

Table 2 Results of damage detection of the 10-element beam using CSS and PSOPC algorithms

Elementnumber

Scenario 1 Scenario 2

CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0000 0 0000 0101 01 0105

2 0000 0 0000 0003 0 0000

3 0000 0 0000 0001 0 0000

4 0000 0 0000 0000 0 0000

5 0000 0 0000 0185 02 0169

6 0100 01 0100 0016 0 0037

7 0000 0 0000 0349 035 0348

8 0000 0 0000 0002 0 0000

9 0000 0 0000 0001 0 0000

10 0000 0 0000 0000 0 0000

Error 00002 00000 00385 00752

Table 3 The values of objective function of the 10-element beam for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 34 times 10

minus0652 times 10

minus0628 times 10

minus0627 times 10

minus0683 times 10

minus0665 times 10

minus0641 times 10

minus0638 times 10

minus0619 times 10

minus0644 times 10

minus06 100PSOPC 25 times 10

minus0713 times 10

minus0799 times 10

minus0814 times 10

minus0721 times 10

minus0746 times 10

minus0726 times 10

minus0716 times 10

minus0688 times 10

minus0811 times 10

minus07 100Scenario 2

CSS 00008 00001 00009 00004 00006 00002 00005 00008 00009 00006 80PSOPC 00003 00013 00056 00004 00074 00074 00080 00011 00006 00085 40

Table 4 Cross-sectional areas of truss elements

Member Area (cm2)1ndash6 187ndash12 1513ndash17 1018ndash25 12

Table 5 Simulated damage scenarios in Bowstring truss

Element number Scenario 1 Scenario 235 108 251011 4024 20

where 119889119894119860 and 119889119894119878 are the actual and estimated damage of the

119894th element using the presented method respectivelyIn Table 3 the values of the objective function for CSS

and PSOPC algorithms in independent 10 runs are given theconvergence history for the 10-element beam in scenarios 1and 2 is shown in Figure 3

It is seen in Table 2 that in the first scenario where thenumber of damaged elements is low the damage identifi-cation is well by both algorithms in the second scenariothe algorithm PSOPC has some wrong in damage detectionso that there is difference between the obtained damage viathis algorithm and the actual intensity of damage in thefifth element and the undamaged sixth element mistakenlyhas been detected to be damaged Meanwhile the locationand amount of damage in structure are obtained by the CSSalgorithm with acceptable accuracy In order to investigatethe noise effects on the results of the CSSmethod 015 noisein measurement is considered for the natural frequencies

Diagrams of the CSS algorithmrsquos damage detection withcomplete and incomplete dynamic noisy data are plotted inFigures 4 and 5

As the results show this method is able to detect thelocation andmagnitude of damaged elements in all scenariosusing complete and incomplete noisy data

Example 2 A Bowstring plane truss with 25 elements isshown in Figure 6 The properties of members of this struc-ture are 119864 = 207 times 0

9Nm2 and 120588 = 7860 kgm3 and cross-sectional areas of elements are given in Table 4

Three scenarios for this truss are considered according toTable 5

Shock and Vibration 9

Table 6 Results of damage detection of the 25-bar Bowstring truss using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0009 0 0000 0000 0 00002 0000 0 0000 0000 0 00003 0000 0 0000 0000 0 00004 0000 0 0000 0000 0 00005 0000 0 0000 0098 01 00956 0000 0 0000 0000 0 00007 0000 0 0000 0000 0 00008 0246 025 0250 0000 0 00009 0001 0 0000 0000 0 000010 0000 0 0000 0000 0 000011 0000 0 0000 0402 04 042612 0000 0 0000 0000 0 000013 0000 0 0000 0000 0 000014 0001 0 0000 0000 0 000015 0000 0 0000 0000 0 000016 0000 0 0000 0000 0 000017 0000 0 0000 0000 0 000018 0000 0 0000 0005 0 000019 0000 0 0000 0004 0 007520 0000 0 0000 0001 0 000021 0000 0 0000 0000 0 000022 0000 0 0000 0000 0 000023 0000 0 0000 0000 0 000024 0000 0 0000 0185 02 000025 0000 0 0000 0014 0 0265Error 00085 00000 00119 01609

In Table 6 the results of two algorithms in all scenarios arecompared Only the first ten natural frequencies are used fordamage detection

The comparison of the objective function for CSS andPSOPC algorithms is made in Table 7 and the convergencehistory for the Bowstring truss in scenarios 1 and 2 is shownin Figure 7

In Figures 8 and 9 the results of CSS algorithm in damagedetection of this truss that is affected by noise are shown

Table 7 and Figures 7 8 and 9 show better results indamage detection by CSS algorithm rather than PSOPCalgorithm and also by applying noise in structures

Example 3 A three-story three-bay frame as shown inFigure 10 is used to verify the damage detection methodexplained in this paper The number of elements and nodesis 39 and 34 respectively

For unbraced plane frame problem all columns areW14 times 132 (119860 = 0025m2 (388 in2) and 119868 = 6386 times

10minus4m4 (1530 in4)) and all beams are W12 times 65 Youngrsquos

modulus is 119864 = 207GPa (30 000 ksi) Poissonrsquos ratio is ] =

03 and the mass density is 120588 = 7780 kgm3 (0000728 lb minuss2in4)

In this example two scenarios from the aspect of elementnumber and its level of damage were assumed that are givenin Table 8 Between ninety natural frequencies only the firsttwenty natural frequencies are utilized for damage detectionin the first scenario

The severity of damage detection by PSOPC and CSSalgorithms in different members of the frame and in eachscenario is shown separately in Table 9 In Table 10 theobtained objective function for both algorithms in each runis given also the convergence history is shown in Figure 11

Also the results of the CSS algorithm damage detectionwith noisy data are plotted in the diagrams of Figures 12 and13

The results of damage detection with two algorithmsin this large frame show that in case of only one memberdamaged there is more accuracy with PSOPC algorithm inidentifying of desired member and obtaining the amount ofdamage than the CSS algorithm It is observed that increasingthe number of injured elements in the structures reducesthe result precision with algorithms From results errors itcan be concluded that the PSOPC algorithm has problemin detecting the location of some of the damaged structuralelements so that in the third scenario the tenth and twenty-fourth members which are injured are diagnosed as healthy

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

International Journal of

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VLSI Design

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Shock and Vibration

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Advances inOptoElectronics

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

Shock and Vibration 7

1

2

13

23 24

14 16 181715

22 25 26 27

19 2120

28 3029

32 3331 34 35 36 37 3938

3 6 9 12

5 8 11

4 7 104m

35m

8m 7m 9m

35m

W12 times 65

W14 times 132

Figure 10 Geometry of unbraced plane frame

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

01

02

03

04

05

Number of analyses

Obj

ectiv

e fun

ctio

n

(a) Scenario 1

CSSPSOPC

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Number of analyses

0

005

01

015

02

025

03

035

04

Obj

ectiv

e fun

ctio

n

(b) Scenario 2

Figure 11 The convergence history of the 39-element frame for the CSS and PSOPC algorithms

The algorithm has been implemented in the commercialMATLAB software and because of the stochastic nature of thealgorithms each example is independently solved ten times

Damage values have been limited to a region between 0and 1

Example 1 A cantilever steel beam is shown in Figure 2 Forthe reason of modal analyzing the beam was divided into 10two-dimensional beam elements with 20 degrees of freedom

The section of the beam is W12 times 65 with mechanicalproperties of

119860 = 00123 m2 (191 in2)

119868 = 2218 times 10minus4 m4 (533 in4)

119864 = 207 times 109Nm2

120588 = 7860 kgm3

The measured modal responses of the beam before and afterdamage were created using the proposed forward solver(for each scenario) Instead of experimental measurementsnumerically generated measurements were used to estimatethe proposed inverse procedure In this example the first tennatural frequencies are used for damage detection Two dif-ferent simulated damage scenarios are considered (Table 1)

These damage scenarios are considered to represent theeffect of severity of damage (stiffness reduction) numberof damaged elements and contribution of damage elementson the results The results of damage detecting of CSS andPSOPC algorithms are given in Table 2 In order to evaluatethe performance of the methods used the summation ofestimated error of damage detection was calculated (Table 2)that is presented as follows

Error = sum(119889119894119860 minus 119889119894119878) (15)

8 Shock and Vibration

Table 2 Results of damage detection of the 10-element beam using CSS and PSOPC algorithms

Elementnumber

Scenario 1 Scenario 2

CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0000 0 0000 0101 01 0105

2 0000 0 0000 0003 0 0000

3 0000 0 0000 0001 0 0000

4 0000 0 0000 0000 0 0000

5 0000 0 0000 0185 02 0169

6 0100 01 0100 0016 0 0037

7 0000 0 0000 0349 035 0348

8 0000 0 0000 0002 0 0000

9 0000 0 0000 0001 0 0000

10 0000 0 0000 0000 0 0000

Error 00002 00000 00385 00752

Table 3 The values of objective function of the 10-element beam for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 34 times 10

minus0652 times 10

minus0628 times 10

minus0627 times 10

minus0683 times 10

minus0665 times 10

minus0641 times 10

minus0638 times 10

minus0619 times 10

minus0644 times 10

minus06 100PSOPC 25 times 10

minus0713 times 10

minus0799 times 10

minus0814 times 10

minus0721 times 10

minus0746 times 10

minus0726 times 10

minus0716 times 10

minus0688 times 10

minus0811 times 10

minus07 100Scenario 2

CSS 00008 00001 00009 00004 00006 00002 00005 00008 00009 00006 80PSOPC 00003 00013 00056 00004 00074 00074 00080 00011 00006 00085 40

Table 4 Cross-sectional areas of truss elements

Member Area (cm2)1ndash6 187ndash12 1513ndash17 1018ndash25 12

Table 5 Simulated damage scenarios in Bowstring truss

Element number Scenario 1 Scenario 235 108 251011 4024 20

where 119889119894119860 and 119889119894119878 are the actual and estimated damage of the

119894th element using the presented method respectivelyIn Table 3 the values of the objective function for CSS

and PSOPC algorithms in independent 10 runs are given theconvergence history for the 10-element beam in scenarios 1and 2 is shown in Figure 3

It is seen in Table 2 that in the first scenario where thenumber of damaged elements is low the damage identifi-cation is well by both algorithms in the second scenariothe algorithm PSOPC has some wrong in damage detectionso that there is difference between the obtained damage viathis algorithm and the actual intensity of damage in thefifth element and the undamaged sixth element mistakenlyhas been detected to be damaged Meanwhile the locationand amount of damage in structure are obtained by the CSSalgorithm with acceptable accuracy In order to investigatethe noise effects on the results of the CSSmethod 015 noisein measurement is considered for the natural frequencies

Diagrams of the CSS algorithmrsquos damage detection withcomplete and incomplete dynamic noisy data are plotted inFigures 4 and 5

As the results show this method is able to detect thelocation andmagnitude of damaged elements in all scenariosusing complete and incomplete noisy data

Example 2 A Bowstring plane truss with 25 elements isshown in Figure 6 The properties of members of this struc-ture are 119864 = 207 times 0

9Nm2 and 120588 = 7860 kgm3 and cross-sectional areas of elements are given in Table 4

Three scenarios for this truss are considered according toTable 5

Shock and Vibration 9

Table 6 Results of damage detection of the 25-bar Bowstring truss using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0009 0 0000 0000 0 00002 0000 0 0000 0000 0 00003 0000 0 0000 0000 0 00004 0000 0 0000 0000 0 00005 0000 0 0000 0098 01 00956 0000 0 0000 0000 0 00007 0000 0 0000 0000 0 00008 0246 025 0250 0000 0 00009 0001 0 0000 0000 0 000010 0000 0 0000 0000 0 000011 0000 0 0000 0402 04 042612 0000 0 0000 0000 0 000013 0000 0 0000 0000 0 000014 0001 0 0000 0000 0 000015 0000 0 0000 0000 0 000016 0000 0 0000 0000 0 000017 0000 0 0000 0000 0 000018 0000 0 0000 0005 0 000019 0000 0 0000 0004 0 007520 0000 0 0000 0001 0 000021 0000 0 0000 0000 0 000022 0000 0 0000 0000 0 000023 0000 0 0000 0000 0 000024 0000 0 0000 0185 02 000025 0000 0 0000 0014 0 0265Error 00085 00000 00119 01609

In Table 6 the results of two algorithms in all scenarios arecompared Only the first ten natural frequencies are used fordamage detection

The comparison of the objective function for CSS andPSOPC algorithms is made in Table 7 and the convergencehistory for the Bowstring truss in scenarios 1 and 2 is shownin Figure 7

In Figures 8 and 9 the results of CSS algorithm in damagedetection of this truss that is affected by noise are shown

Table 7 and Figures 7 8 and 9 show better results indamage detection by CSS algorithm rather than PSOPCalgorithm and also by applying noise in structures

Example 3 A three-story three-bay frame as shown inFigure 10 is used to verify the damage detection methodexplained in this paper The number of elements and nodesis 39 and 34 respectively

For unbraced plane frame problem all columns areW14 times 132 (119860 = 0025m2 (388 in2) and 119868 = 6386 times

10minus4m4 (1530 in4)) and all beams are W12 times 65 Youngrsquos

modulus is 119864 = 207GPa (30 000 ksi) Poissonrsquos ratio is ] =

03 and the mass density is 120588 = 7780 kgm3 (0000728 lb minuss2in4)

In this example two scenarios from the aspect of elementnumber and its level of damage were assumed that are givenin Table 8 Between ninety natural frequencies only the firsttwenty natural frequencies are utilized for damage detectionin the first scenario

The severity of damage detection by PSOPC and CSSalgorithms in different members of the frame and in eachscenario is shown separately in Table 9 In Table 10 theobtained objective function for both algorithms in each runis given also the convergence history is shown in Figure 11

Also the results of the CSS algorithm damage detectionwith noisy data are plotted in the diagrams of Figures 12 and13

The results of damage detection with two algorithmsin this large frame show that in case of only one memberdamaged there is more accuracy with PSOPC algorithm inidentifying of desired member and obtaining the amount ofdamage than the CSS algorithm It is observed that increasingthe number of injured elements in the structures reducesthe result precision with algorithms From results errors itcan be concluded that the PSOPC algorithm has problemin detecting the location of some of the damaged structuralelements so that in the third scenario the tenth and twenty-fourth members which are injured are diagnosed as healthy

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

Table 2 Results of damage detection of the 10-element beam using CSS and PSOPC algorithms

Elementnumber

Scenario 1 Scenario 2

CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0000 0 0000 0101 01 0105

2 0000 0 0000 0003 0 0000

3 0000 0 0000 0001 0 0000

4 0000 0 0000 0000 0 0000

5 0000 0 0000 0185 02 0169

6 0100 01 0100 0016 0 0037

7 0000 0 0000 0349 035 0348

8 0000 0 0000 0002 0 0000

9 0000 0 0000 0001 0 0000

10 0000 0 0000 0000 0 0000

Error 00002 00000 00385 00752

Table 3 The values of objective function of the 10-element beam for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 34 times 10

minus0652 times 10

minus0628 times 10

minus0627 times 10

minus0683 times 10

minus0665 times 10

minus0641 times 10

minus0638 times 10

minus0619 times 10

minus0644 times 10

minus06 100PSOPC 25 times 10

minus0713 times 10

minus0799 times 10

minus0814 times 10

minus0721 times 10

minus0746 times 10

minus0726 times 10

minus0716 times 10

minus0688 times 10

minus0811 times 10

minus07 100Scenario 2

CSS 00008 00001 00009 00004 00006 00002 00005 00008 00009 00006 80PSOPC 00003 00013 00056 00004 00074 00074 00080 00011 00006 00085 40

Table 4 Cross-sectional areas of truss elements

Member Area (cm2)1ndash6 187ndash12 1513ndash17 1018ndash25 12

Table 5 Simulated damage scenarios in Bowstring truss

Element number Scenario 1 Scenario 235 108 251011 4024 20

where 119889119894119860 and 119889119894119878 are the actual and estimated damage of the

119894th element using the presented method respectivelyIn Table 3 the values of the objective function for CSS

and PSOPC algorithms in independent 10 runs are given theconvergence history for the 10-element beam in scenarios 1and 2 is shown in Figure 3

It is seen in Table 2 that in the first scenario where thenumber of damaged elements is low the damage identifi-cation is well by both algorithms in the second scenariothe algorithm PSOPC has some wrong in damage detectionso that there is difference between the obtained damage viathis algorithm and the actual intensity of damage in thefifth element and the undamaged sixth element mistakenlyhas been detected to be damaged Meanwhile the locationand amount of damage in structure are obtained by the CSSalgorithm with acceptable accuracy In order to investigatethe noise effects on the results of the CSSmethod 015 noisein measurement is considered for the natural frequencies

Diagrams of the CSS algorithmrsquos damage detection withcomplete and incomplete dynamic noisy data are plotted inFigures 4 and 5

As the results show this method is able to detect thelocation andmagnitude of damaged elements in all scenariosusing complete and incomplete noisy data

Example 2 A Bowstring plane truss with 25 elements isshown in Figure 6 The properties of members of this struc-ture are 119864 = 207 times 0

9Nm2 and 120588 = 7860 kgm3 and cross-sectional areas of elements are given in Table 4

Three scenarios for this truss are considered according toTable 5

Shock and Vibration 9

Table 6 Results of damage detection of the 25-bar Bowstring truss using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0009 0 0000 0000 0 00002 0000 0 0000 0000 0 00003 0000 0 0000 0000 0 00004 0000 0 0000 0000 0 00005 0000 0 0000 0098 01 00956 0000 0 0000 0000 0 00007 0000 0 0000 0000 0 00008 0246 025 0250 0000 0 00009 0001 0 0000 0000 0 000010 0000 0 0000 0000 0 000011 0000 0 0000 0402 04 042612 0000 0 0000 0000 0 000013 0000 0 0000 0000 0 000014 0001 0 0000 0000 0 000015 0000 0 0000 0000 0 000016 0000 0 0000 0000 0 000017 0000 0 0000 0000 0 000018 0000 0 0000 0005 0 000019 0000 0 0000 0004 0 007520 0000 0 0000 0001 0 000021 0000 0 0000 0000 0 000022 0000 0 0000 0000 0 000023 0000 0 0000 0000 0 000024 0000 0 0000 0185 02 000025 0000 0 0000 0014 0 0265Error 00085 00000 00119 01609

In Table 6 the results of two algorithms in all scenarios arecompared Only the first ten natural frequencies are used fordamage detection

The comparison of the objective function for CSS andPSOPC algorithms is made in Table 7 and the convergencehistory for the Bowstring truss in scenarios 1 and 2 is shownin Figure 7

In Figures 8 and 9 the results of CSS algorithm in damagedetection of this truss that is affected by noise are shown

Table 7 and Figures 7 8 and 9 show better results indamage detection by CSS algorithm rather than PSOPCalgorithm and also by applying noise in structures

Example 3 A three-story three-bay frame as shown inFigure 10 is used to verify the damage detection methodexplained in this paper The number of elements and nodesis 39 and 34 respectively

For unbraced plane frame problem all columns areW14 times 132 (119860 = 0025m2 (388 in2) and 119868 = 6386 times

10minus4m4 (1530 in4)) and all beams are W12 times 65 Youngrsquos

modulus is 119864 = 207GPa (30 000 ksi) Poissonrsquos ratio is ] =

03 and the mass density is 120588 = 7780 kgm3 (0000728 lb minuss2in4)

In this example two scenarios from the aspect of elementnumber and its level of damage were assumed that are givenin Table 8 Between ninety natural frequencies only the firsttwenty natural frequencies are utilized for damage detectionin the first scenario

The severity of damage detection by PSOPC and CSSalgorithms in different members of the frame and in eachscenario is shown separately in Table 9 In Table 10 theobtained objective function for both algorithms in each runis given also the convergence history is shown in Figure 11

Also the results of the CSS algorithm damage detectionwith noisy data are plotted in the diagrams of Figures 12 and13

The results of damage detection with two algorithmsin this large frame show that in case of only one memberdamaged there is more accuracy with PSOPC algorithm inidentifying of desired member and obtaining the amount ofdamage than the CSS algorithm It is observed that increasingthe number of injured elements in the structures reducesthe result precision with algorithms From results errors itcan be concluded that the PSOPC algorithm has problemin detecting the location of some of the damaged structuralelements so that in the third scenario the tenth and twenty-fourth members which are injured are diagnosed as healthy

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

Table 6 Results of damage detection of the 25-bar Bowstring truss using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0009 0 0000 0000 0 00002 0000 0 0000 0000 0 00003 0000 0 0000 0000 0 00004 0000 0 0000 0000 0 00005 0000 0 0000 0098 01 00956 0000 0 0000 0000 0 00007 0000 0 0000 0000 0 00008 0246 025 0250 0000 0 00009 0001 0 0000 0000 0 000010 0000 0 0000 0000 0 000011 0000 0 0000 0402 04 042612 0000 0 0000 0000 0 000013 0000 0 0000 0000 0 000014 0001 0 0000 0000 0 000015 0000 0 0000 0000 0 000016 0000 0 0000 0000 0 000017 0000 0 0000 0000 0 000018 0000 0 0000 0005 0 000019 0000 0 0000 0004 0 007520 0000 0 0000 0001 0 000021 0000 0 0000 0000 0 000022 0000 0 0000 0000 0 000023 0000 0 0000 0000 0 000024 0000 0 0000 0185 02 000025 0000 0 0000 0014 0 0265Error 00085 00000 00119 01609

In Table 6 the results of two algorithms in all scenarios arecompared Only the first ten natural frequencies are used fordamage detection

The comparison of the objective function for CSS andPSOPC algorithms is made in Table 7 and the convergencehistory for the Bowstring truss in scenarios 1 and 2 is shownin Figure 7

In Figures 8 and 9 the results of CSS algorithm in damagedetection of this truss that is affected by noise are shown

Table 7 and Figures 7 8 and 9 show better results indamage detection by CSS algorithm rather than PSOPCalgorithm and also by applying noise in structures

Example 3 A three-story three-bay frame as shown inFigure 10 is used to verify the damage detection methodexplained in this paper The number of elements and nodesis 39 and 34 respectively

For unbraced plane frame problem all columns areW14 times 132 (119860 = 0025m2 (388 in2) and 119868 = 6386 times

10minus4m4 (1530 in4)) and all beams are W12 times 65 Youngrsquos

modulus is 119864 = 207GPa (30 000 ksi) Poissonrsquos ratio is ] =

03 and the mass density is 120588 = 7780 kgm3 (0000728 lb minuss2in4)

In this example two scenarios from the aspect of elementnumber and its level of damage were assumed that are givenin Table 8 Between ninety natural frequencies only the firsttwenty natural frequencies are utilized for damage detectionin the first scenario

The severity of damage detection by PSOPC and CSSalgorithms in different members of the frame and in eachscenario is shown separately in Table 9 In Table 10 theobtained objective function for both algorithms in each runis given also the convergence history is shown in Figure 11

Also the results of the CSS algorithm damage detectionwith noisy data are plotted in the diagrams of Figures 12 and13

The results of damage detection with two algorithmsin this large frame show that in case of only one memberdamaged there is more accuracy with PSOPC algorithm inidentifying of desired member and obtaining the amount ofdamage than the CSS algorithm It is observed that increasingthe number of injured elements in the structures reducesthe result precision with algorithms From results errors itcan be concluded that the PSOPC algorithm has problemin detecting the location of some of the damaged structuralelements so that in the third scenario the tenth and twenty-fourth members which are injured are diagnosed as healthy

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

Table 7 The values of objective function of the Bowstring truss for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 32 times 10

minus0543 times 10

minus0549 times 10

minus0569 times 10

minus0571 times 10

minus0533 times 10

minus0526 times 10

minus0575 times 10

minus0534 times 10

minus0526 times 10

minus05 100PSOPC 26 times 10

minus0749 times 10

minus0218 times 10

minus0713 times 10

minus0112 times 10

minus0124 times 10

minus0850 times 10

minus0819 times 10

minus0826 times 10

minus0217 times 10

minus02 60Scenario 2

CSS 00002 00003 00003 00001 00039 00038 00002 00003 00015 00051 60PSOPC 00005 00134 00672 00629 00598 00050 00049 00065 00123 00251 10

Incomplete noisy dataComplete noisy dataActual damage

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 390

5

10

15

20

25

30

35

40

Element number

Dam

age (

)

Figure 12 Damage distribution of plane frame using the completeand incomplete noisy data in Scenario 1

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 400

10

20

30

40

50

60

Element number

Dam

age (

)

Predicted damageActual damage

Figure 13 Damage distribution of plane frame using the completenoisy data in Scenario 2

Table 8 Simulated damage scenarios in unbraced plane frame

Element number Scenario 1 Scenario 21 4045 309 1010 1013 2014 4019 2524 2032 50

and a considerable amount of damage in the fifteenth andtwenty-third elements that have no damage has been gainedHowever the CSS algorithm has well identified places ofall damaged elements in the structure and has achieved theamount of their damage with high accuracy and low errorFrom Figures 12 and 13 it can be observed that even in thatcase a lot of damage to the frame is considered the effectof noise on the results obtained by the CSS algorithm islow so that in case of incomplete data accuracy of resultshas reduced slightly and the performance of this algorithmdespite the noise was considered acceptable

In all three examples the convergences diagrams areshown that the process of convergence is gradual So it causesa more comprehensive search algorithm for finding optimalsolutions

The result increases the success rate of this type ofalgorithm performance in comparison with the PSOPCalgorithm

6 Conclusions

An approach for detecting damage based on continuumdamage model using charged system search algorithm ispresented The algorithm evaluates the location and severityof damage in three structures a cantilever beam a Bowstringplane truss and a three-story three-bay unbraced plane frameby minimizing an objective function by measuring completeand incomplete noisy modal data with different damagescenarios

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

Table 9 Results of damage detection of the 39-element frame using CSS and PSOPC algorithms

Element number Scenario 1 Scenario 2CSS Actual damage PSOPC CSS Actual damage PSOPC

1 0399 0400 0393 0005 0 00002 0001 0 0000 0001 0 00003 0001 0 0000 0000 0 00004 0001 0 0027 0037 0 00005 0003 0 0000 0254 0300 03566 0000 0 0000 0016 0 00007 0000 0 0000 0013 0 00008 0005 0 0000 0001 0 00009 0056 0100 0000 0001 0 000010 0002 0 0000 0095 0100 000011 0005 0 0000 0002 0 000012 0008 0 0000 0001 0 000013 0193 0200 0195 0058 0 000014 0001 0 0000 0374 0400 034915 0004 0 0000 0005 0 011516 0000 0 0000 0001 0 000017 0000 0 0000 0001 0 000018 0000 0 0000 0004 0 000019 0000 0 0000 0212 0250 025620 0000 0 0000 0004 0 000021 0000 0 0000 0029 0 000022 0000 0 0002 0015 0 000023 0000 0 0000 0001 0 009224 0000 0 0000 0174 0200 000025 0000 0 0000 0001 0 000026 0000 0 0000 0002 0 000027 0002 0 0000 0005 0 000028 0000 0 0000 0001 0 004029 0000 0 0001 0007 0 000030 0000 0 0000 0000 0 000031 0001 0 0003 0002 0 000032 0000 0 0000 0497 0500 049533 0000 0 0000 0000 0 000034 0006 0 0000 0000 0 000035 0005 0 0025 0001 0 000036 0006 0 0000 0001 0 000037 0002 0 0000 0001 0 001538 0007 0 0000 0001 0 000039 0003 0 0000 0002 0 0000Error 01162 01700 03600 06808

In order to demonstrate the power of this algorithm in thediagnosis of damage a comparison has been made betweenthe results of this algorithm and the PSOPC algorithm

By comparing the damage detection results of the twovarious methods some interesting points have been con-cluded In scenarios where the number of damaged elementsis considered low both algorithms have acceptable accuracyin the results considering the more damaged members

of structures the PSOPC algorithm has been mistaken toidentify the damaged elements of the structureThis problemin larger-scale structures and the increasing number ofmembers has beenmore visible CSS algorithm has identifiedthe exact location of damage as well as achieving the amountof damage with acceptable accuracy It can be concluded thatthe CSS algorithm has great potential in global and localsearch of damage

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Shock and Vibration

Table 10 The values of objective function of the 39-element frame for algorithms

Run number1 2 3 4 5 6 7 8 9 10 Success

Scenario 1CSS 44 times 10

minus0346 times 10

minus0387 times 10

minus0462 times 10

minus0411 times 10

minus0366 times 10

minus0437 times 10

minus0480 times 10

minus0429 times 10

minus0398 times 10

minus04 50PSOPC 12 times 10

minus0213 times 10

minus0244 times 10

minus0315 times 10

minus0243 times 10

minus0213 times 10

minus0339 times 10

minus0214 times 10

minus0239 times 10

minus0315 times 10

minus02 10Scenario 2

CSS 19 times 10minus03

13 times 10minus03

17 times 10minus03

12 times 10minus03

23 times 10minus03

28 times 10minus03

19 times 10minus03

11 times 10minus03

21 times 10minus03

16 times 10minus03 30

PSOPC 29 times 10minus02

39 times 10minus02

11 times 10minus02

18 times 10minus02

11 times 10minus02

59 times 10minus03

69 times 10minus02

36 times 10minus02

53 times 10minus02

13 times 10minus02 0

Due to noise the real value of natural frequencies ofstructure will change it can be seen from the diagramsthat in these cases accuracy reduction of the results of theCSS algorithm is very low This indicates the high powerof this algorithm in damage detection considering noise instructure

Charged system search by considering small populationand low iteration cycle can detect the severity and location ofdamage with acceptable accuracy which shows high powerand speed of convergence of this algorithm The proposedmethod by the charged system search algorithm producedbetter results compared with particle swarm optimizationmethod

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] D J Ewins Modal Testing Theory and Practice John Wiley ampSons New York NY USA 1984

[2] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[3] N Hu XWang H Fukunaga Z H Yao H X Zhang and Z SWu ldquoDamage assessment of structures using modal test datardquoInternational Journal of Solids and Structures vol 38 no 18 pp3111ndash3126 2001

[4] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[5] N Bicanic and H-P Chen ldquoDamage identification in framedstructures using natural frequenciesrdquo International Journal forNumerical Methods in Engineering vol 40 no 23 pp 4451ndash4468 1997

[6] A K Pandey M Biswas and M M Samman ldquoDamagedetection from changes in curvature mode shapesrdquo Journal ofSound and Vibration vol 145 no 2 pp 321ndash332 1991

[7] J E Mottershead and M I Friswell ldquoModel updating instructural dynamics a surveyrdquo Journal of Sound and Vibrationvol 167 no 2 pp 347ndash375 1993

[8] R Perera andR Torres ldquoStructural damage detection viamodaldata with genetic algorithmsrdquo Journal of Structural Engineeringvol 132 no 9 pp 1491ndash1501 2006

[9] S Fallahian and S M Seyedpoor ldquoA two stage method forstructural damage identification using an adaptive neuro-fuzzyinference system and particle Swarm optimizationrdquo AsianJournal of Civil Engineering (Building and Housing) vol 11 pp797ndash810 2010

[10] B H Koh and S J Dyke ldquoStructural health monitoring forflexible bridge structures using correlation and sensitivity ofmodal datardquo Computers and Structures vol 85 no 3-4 pp 117ndash130 2007

[11] R-S He and S-F Hwang ldquoDamage detection by a hybrid real-parameter genetic algorithm under the assistance of grey rela-tion analysisrdquo Engineering Applications of Artificial Intelligencevol 20 no 7 pp 980ndash992 2007

[12] H Y Guo and Z L Li ldquoA two-stage method to identifystructural damage sites and extents by using evidence theoryand micro-search genetic algorithmrdquo Mechanical Systems andSignal Processing vol 23 no 3 pp 769ndash782 2009

[13] L Yu and X Chen ldquoBridge damage identification by combiningmodal flexibility and PSO methodsrdquo in Proceedings of thePrognostics and System Health Management Conference (PHMrsquo10) Macau China January 2010

[14] Z Tabrizian E Afshari G Ghodrati Amiri M H A Beigyand SM PourhoseiniNejad ldquoAnewdamage detectionmethodBig Bang-Big Crunch (BB-BC) algorithmrdquo Journal of Shock andVibration vol 20 no 4 pp 633ndash648 2013

[15] A Kaveh and S Talatahari ldquoA novel heuristic optimizationmethod charged system searchrdquo Acta Mechanica vol 213 no3-4 pp 267ndash289 2010

[16] S M Pourhoseini Nejad G Ghodrati Amiri A Asadi EAfshari and Z Tabrizian ldquoDamage detection of skeletal struc-tures using particle swarm optimizer with passive congregation(PSOPC) algorithm via incomplete modal datardquo Journal ofComputational Methods in Civil Engineering vol 3 no 1 pp1ndash13 2012

[17] S He Q H Wu J Y Wen J R Saunders and R CPaton ldquoA particle swarm optimizer with passive congregationrdquoBioSystems vol 78 no 1ndash3 pp 135ndash147 2004

[18] X Wang N Hu H Fukunaga and Z H Yao ldquoStructuraldamage identification using static test data and changes infrequenciesrdquo Engineering Structures vol 23 no 6 pp 610ndash6212001

[19] A Messina E J Williams and T Contursi ldquoStructural damagedetection by a sensitivity and statistical-based methodrdquo Journalof Sound and Vibration vol 216 no 5 pp 791ndash808 1998

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural and

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

mechanical systems from changes in their vibration character-istics a literature reviewrdquo Research Report LA-13070-MS ESA-EA Los Alamos National Laboratory Los Alamos NM USA1996

[21] M Nobahari and S M Seyedpoor ldquoStructural damage detec-tion using an efficient correlation-based index and a modifiedgenetic algorithmrdquoMathematical and Computer Modelling vol53 no 9-10 pp 1798ndash1809 2011

[22] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[23] A Kaveh and S Talatahari ldquoParticle swarm optimizer antcolony strategy and harmony search scheme hybridized foroptimization of truss structuresrdquoComputers and Structures vol87 no 5-6 pp 267ndash283 2009

[24] H J Salane and J W Baldwin Jr ldquoIdentification of modalproperties of bridgesrdquo Journal of Structural Engineering vol 116no 7 pp 2008ndash2021 1990

[25] O S Salawu and CWilliams ldquoBridge assessment using forced-vibration testingrdquo Journal of Structural Engineering vol 121 no2 pp 161ndash173 1995

[26] J-M Ndambi J Vantomme and K Harri ldquoDamage assessmentin reinforced concrete beams using eigenfrequencies and modeshape derivativesrdquo Engineering Structures vol 24 no 4 pp 501ndash515 2002

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of