Research ArticleA New Flexibility Based Damage Index forDamage Detection of Truss Structures

M Montazer and S M Seyedpoor

Department of Civil Engineering Shomal University Amol 46161-84596 Iran

Correspondence should be addressed to S M Seyedpoor smseyedpoorgmailcom

Received 19 July 2013 Accepted 15 November 2013 Published 16 March 2014

Academic Editor Gyuhae Park

Copyright copy 2014 M Montazer and S M Seyedpoor This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A new damage index called strain change based on flexibility index (SCBFI) is introduced to locate damaged elements of trusssystems The principle of SCBFI is based on considering strain changes in structural elements between undamaged and damagedstates The strain of an element is evaluated using the columnar coefficients of the flexibility matrix estimated via modal analysisinformation Two illustrative test examples are considered to assess the performance of the proposed method Numerical resultsindicate that the method can provide a reliable tool to accurately identify the multiple-structural damage for truss structures

1 Introduction

Structural damage detection has a great importance in civilengineering Neglecting the local damage may cause thereduction of the functional age of a structural system oreven an overall failure of the structure Therefore damagedetection is an important issue in structural engineeringThebasis of many damage identification procedures is observ-ing the changes in structural responses Damage reducesstructurersquos stiffness and mass which leads to a change inthe static and dynamic responses of the structure Thereforethe damage detection techniques are generally classified intotwo main categories They include the dynamic and staticidentification methods requiring the dynamic and static testdata respectively Because of the global nature of the dynamicresponses of a structure techniques for detecting damagebased on vibration characteristics of structures have beengaining importance

Presence of a crack or localized damage in a structurereduces its stiffness leading to the decrease of the naturalfrequencies and the change of vibration modes of the struc-ture [1ndash3] Many researchers have used one or more of thesecharacteristics to detect and locate the structural damageCawley and Adams [4] used the changes in the naturalfrequencies together with a finite element model to locate thedamage site Although it is fairly easy to detect the presence

of damage in a structure from changes in the natural frequen-cies it is difficult to determine the location of damage Thisis because damage at two different locations associated witha certain amount of damage may produce the same amountof frequency change Furthermore in the case of symmetricstructures the changes in the natural frequencies due to dam-age at two symmetric locations are exactly the same Thereis thus a need for a more comprehensive method of damageassessment in structures To overcome this drawback modeshapes have been used for identifying the damage location[5 6] Displacement mode shapes can be obtained by exper-imental tools but an accurate characterization of the damagelocation from these parameters requires measurements inmany locationsTherefore using the changes in mode shapesbetween damaged and undamaged structures may not bevery efficientMannan and Richardson utilized the frequencyresponse function (FRF) measurements for detecting andlocating the structural cracks [7] Pandey and Biswas used thechange in flexibility matrix to detect damage in a beam [8]Raghavendrachar and Aktan calculated the flexibility matrixbased on mode shapes and demonstrated the advantagesof using the flexibility compared to the mode shapes [9]The influence of statistical errors on damage detection basedon structural flexibility and mode shape curvature has beeninvestigated by Tomaszewska [10] The generalized flexibilitymatrix change to detect the location and extent of structural

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 460692 12 pageshttpdxdoiorg1011552014460692

2 Shock and Vibration

damage has been introduced by Li et al [11] Compared withthe original flexibility matrix based approach the effect oftruncating higher-order modes can be considerably reducedin the new approach A method based on best achievableflexibility change with capability in detecting the locationand extent of structural damage has also been presentedby Yang and Sun [12] A method for structural damageidentification based on flexibility disassembly has also beenproposed by Yang [13]The efficiency of the proposedmethodhas been demonstrated by numerical examples A flexibilitybased damage index has been proposed for structural damagedetection by Zhang et al [14] Example of a 12-storey buildingmodel illustrated the efficiency of the proposed index forsuccessfully detecting damage locations in both the singleand multiple damage cases The flexibility matrix and strainenergy concepts of a structure have been used by Nobahariand Seyedpoor [15] in order to introduce a damage indicatorfor locating structural damage

In this paper a new index for structural damage detectionis proposed The principle of the index is based on compar-ison of structural elementrsquos strains obtained from two setsrelated to intact anddamaged structureThe calculation of thestrain is based on a flexibility matrix estimated from modalanalysis information Taking advantage of highly convergedflexibility matrix using only few vibration modes related tolow frequencies in the first phase and then determining theelemental strain using the flexibility coefficients develop arobust tool for damage localization of truss structures

2 Structural Damage Detection

In recent years many damage indices have been proposed toidentify structural damage In this paper a number of widelyused indices are first described and then the new proposeddamage index is introduced

21 Modal Assurance Criterion (MAC) and Coordinate ModalAssurance Criterion (COMAC) Based on the basic modalparameters of structures such as natural frequencies damp-ing ratios and mode shapes some coefficients derived fromthese parameters can be useful for damage detection TheMAC and COMAC factors [16] may be mentioned in thiscategory The factors are derived from mode shapes andexpress the correlation between two mode shapes obtainedfrom two sets Sufficient number of degrees of freedom(number of measurement points) are needed here to attaingood accuracy

Let [120593119860] and [120593

119861] be the first and second sets of measured

mode shapes in a matrix form of size 119899 times 119898119860and 119899 times 119898

119861

respectively 119898119860and 119898

119861are the numbers of mode shapes

considered in the respective sets and 119899 is the number ofmeasurement points The MAC factor is then defined for themode shapes 119895 and 119896 as follows

MAC(119895119896)

=

100381610038161003816100381610038161003816sum119899

119894=1[120593119860]119895

119894sdot [120593119861]119896

119894

100381610038161003816100381610038161003816

2

sum119899

119894=1([120593119860]119895

119894)2

sdot sum119899

119894=1([120593119861]119896

119894)2 (1)

where 119895 = 1 119898119860

and 119896 = 1 119898119861 [120593119860]119895

119894and

[120593119861]119896

119894are the 119894th components of the modes [120593

119860]119895 and [120593

119861]119896

respectively The MAC(119895119896)

factor indicates the degree ofcorrelation between the 119895th mode of the first set 119860 and the119896th mode of the second set 119861 The MAC values vary from0 to 1 with 0 for no correlation and 1 for full correlationTherefore if the eigenvectors [120593

119860] and [120593

119861] are identical the

corresponding MAC values will be close to 1 thus indicatingthe full correlation between the two modes The deviation ofthe factors from 1 can be interpreted as a damage indicator ina structure

The COMAC factors are generally used to identify wherethe mode shapes of the structure from two sets of mea-surements do not correlate If the modal displacements ina coordinate 119895 from two sets of measurements are identicalthe COMAC factor is close to 1 for this coordinate A largedeviation from unity can be interpreted as damage indicationin the structure For the coordinate 119901 of a structure using 119898

mode shapes the COMAC factor is defined as follows

COMAC119901=

10038161003816100381610038161003816sum119898

119894=1[120593119860]119901

119894sdot [120593119861]119901

119894

10038161003816100381610038161003816

2

sum119898

119894=1([120593119860]119901

119894)2

sdot sum119898

119894=1([120593119861]119901

119894)2 (2)

In practice only a few mode shapes of a structure canbe measured As a result a method capable of predictingstructural damage that requires a limited number of modeshapes would be more efficient

22 Flexibility Method It has been proved that the presenceof damage in a structure increases its flexibility So anychange observed in the flexibility matrix can be interpretedas a damage indication in the structure and allows one toidentify damage [17] Therefore another class of damageidentificationmethods is based onusing the flexibilitymatrixThe flexibility matrix 119865 is the inverse of the stiffness matrix119870 relating the applied static forces 119891 to resulting structuraldisplacements 119906 as

119906 = [119865] 119891 (3)

The flexibility matrix can be also dynamically measured frommodal analysis data The relationship between the flexibilitymatrix 119865 and the dynamic characteristics of a structure canbe given by [15 17]

119865 = [120593] [Λ]minus1[120593]119879≃

119899119898

sum119894=1

1

1205962119894

120593119894120593119879

119894 (4)

where [120593] is the mode shape matrix [Λ] = diag(11205962) isa diagonal matrix 120596

119894is the 119894th circular frequency 120593

119894is the

mass-normalized 119894th mode shape of the structure and nm isthe number of measured modes

From (4) one can see that the modal contribution to theflexibility matrix decreases as the frequency increases On theother hand the flexibility matrix converges rapidly with theincrease of the values of the frequenciesTherefore from onlya few of lowermodes a good estimate for the flexibilitymatrixcan be achieved

Shock and Vibration 3

The principle of flexibility method is based on a compar-ison of the flexibility matrices from two sets of mode shapesIf 119865ℎand 119865

119889are the flexibility matrices corresponding to

the healthy and damaged states of the structure a flexibilitychange matrix Δ119865 can be defined as the difference of the twomatrices

Δ119865 = 119865119889minus 119865ℎ (5)

For each degree of freedom let 120575119895be the maximum absolute

value of the elements in the corresponding column of Δ119865

120575119895= max 10038161003816100381610038161003816120575119894119895

10038161003816100381610038161003816 (6)

where 120575119894119895are the elements of Δ119865 In order to identify damage

in the structure the quantity 120575119895can be used as a damage

indicator [17]

23 Modal Strain Energy Based Method The methods basedon modal strain energy of a structure have been commonlyused in damage detection [17ndash19] Since the mode shapevectors are equivalent to nodal displacements of a vibratingstructure therefore in each element of the structure strainenergy is stored The strain energy of a structure due tomode shape vectors is usually referred to as modal strainenergy (MSE) and can be considered as a valuable parameterfor damage identification The modal strain energy of ethelement in 119894th mode of the structure can be expressed as [19]

mse119890119894=

1

2120593119890119879

119894119870119890120593119890

119894119894 = 1 2 119899119898 119890 = 1 2 119899119905119890 (7)

where119870119890 is the stiffnessmatrix of 119890th element of the structureand 120593

119890

119894is the vector of corresponding nodal displacements of

element 119890 in mode 119894A normalized form of MSE considering 119899119898 vibrating

modes of the structure proposed in [19] can be given as

mnmse119890 =sum119899119898

119894=1(mse119890119894sum119899119905119890

119890=1mse119890119894)

119899119898 119890 = 1 119899119905119890 (8)

The damage occurrence increases the MSE and consequentlythe efficient parameter mnmse119890 So by determining theparameter mnmse119890 for each element of healthy and damagedstructures an efficient indicator for identifying the damage inthe element can be defined [19]

3 The Proposed Strain Change Based onFlexibility Index (SCBFI)

In this study a new damage detection index based on consid-ering strain changes in a structural element due to damageis developed The new index SCBFI proposed is capable ofidentifying and locating the multiple-structural damage intruss structures The principle of the new index is based onevaluating the changes of strain in structural elements butthe method of computing the strain is completely differentfrom the usual methodsThe nodal displacement vector usedfor computing the strain obtains from the flexibility matrix of

the structure Moreover the flexibility matrix is determinedfrommodal analysis information including mode shapes andnatural frequencies Taking advantage of rapid convergenceof the flexibility matrix in terms of the number of vibrationmodes helps the proposed index to identify the structuraldamage with more efficiency and lower computational cost

As the first step for constructing the proposed damageindex a modal analysis is required to be performed Themodal analysis is a tool to determine the natural frequenciesand mode shapes of a structure [20] It has the mathematicalform of

(119870 minus 1205962

119894119872)120593119894= 0 119894 = 1 2 119899119898 (9)

where 119870 and 119872 are the stiffness and mass matrices of thestructure respectively Also 120596

119894and 120593

119894are the 119894th circular fre-

quency and mode shape vector of the structure respectivelyAt the second step the flexibility matrix of healthy and

damaged structure (FMH FMD) related to the dynamiccharacteristics of the healthy and damaged structure can beapproximated as

FMH ≃

119899119898

sum119894=1

1

(120596ℎ119894)2120593ℎ

119894120593ℎ

119894

119879

FMD ≃

119899119898

sum119894=1

1

(120596119889119894)2120593119889

119894120593119889

119894

119879

(10)

where 120596ℎ119894and 120596

119889

119894are the 119894th circular frequency of healthy and

damaged structure respectively 120593ℎ119894and 120593

119889

119894are the 119894th mode

shape vector of healthy and damaged structure respectivelyand 119899119898 is the number of vibrating modes considered

It can be observed that all components of the modeshapes are required to be measured and it is not a realisticassumption for operational damage detection However foractual use of the suggested method it is not needed tomeasure the full set of mode shapes The mode shapes ofthe damaged structure in partial degrees of freedom arefirst measured and then the incomplete mode shapes areexpanded tomatch all degrees of freedomof the structure by acommon technique such as a dynamic condensation method[21]

Since each column of the flexibility matrix representsthe displacement pattern of the structure associated with aunit force applied at the corresponding DOF of that columntherefore they can be used as nodal displacements to calculatethe strains of structural elements The strain of each elementof a 2-D truss structure can be expressed as [22]

120598 =1

119871[minus cos 120579 minus sin 120579 cos 120579 sin 120579 ]

1198891119909

1198891119910

1198892119909

1198892119910

(11)

where 120598 is the strain of truss element 119871 is the length ofelement 120579 is the angle between local and global coordinatesand 119889

119879= 1198891119909 1198891119910 1198892119909 1198892119910 is the nodal displacement vector

of the element

4 Shock and Vibration

Start

Modal analysis (natural frequencies and mode shapes of

damaged structure)

Calculate the flexibility matrix ofdamaged structure (FMD) using

a few mode shapes

Consider each column of FMD as displacement pattern for damaged structure

Arrange the strain matrix of damaged structure (SMD)

Determine the strain change matrix as

Modal analysis (natural frequencies and mode shapes of

healthy structure)

Calculate the flexibility matrix of healthy structure (FMH) using a

few mode shapes

Consider each column of FMH as displacement pattern for healthy structure

Arrange the strain matrix of healthy structure (SMH)

SCM = SMD minus SMH

Calculate the absolute mean value of each rowof matrix SCM as

SCBFIe = | |sumnci=1 SCM i

e

nc| e =

using (15) and (16)Normalize and scale the SCBFIe

1 2 nte

Figure 1 The process for constructing the SCBFI index

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

152

m

152m

Figure 2 Planar truss having 31 elements

Shock and Vibration 5

001020304

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

SCBF

I

Element number

(a)

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(c)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(d)

Figure 3 Damage identification for 31-bar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 4 Damage identification for 31-bar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

So at the next step the strain of eth element for 119894thcolumn of healthy and damaged structure based on thecolumnar coefficients of the flexibility matrix can be deter-mined as

SMH119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890ℎ

SMD119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890119889

(12)

where SMH119894119890is the strain of 119890th element related to 119894th column

of FMH SMD119894119890is the strain of 119890th element related to 119894th

column of FMD 119889119894119890ℎ and 119889

119894

119890119889 are the nodal displacement

vectors of 119890th element associatedwith the 119894th column of flexi-bility matrix for healthy and damaged structure respectively120579119890is the angle between local and global coordinates of 119890th

element and 119871119890is the length of 119890th element

Now the strain change matrix SCM can be defined as thedifference between the strain matrix of damaged structureSMD and the strain matrix of healthy structure SMH as

SCM = SMD minus SMH (13)

The 119894th rowof SCMmatrix represents the strain changes of 119894thelement of structure for unit loads applied at different DOFsof the structure The SCBFI index can now be defined as acolumnar vector containing the absolute mean value of eachrow of SCMmatrix given by

SCBFI119890=

100381610038161003816100381610038161003816100381610038161003816

sum119899119888

119894=1SCM119894119890

119899119888

100381610038161003816100381610038161003816100381610038161003816 119890 = 1 2 119899119905119890 (14)

where 119899119888 is the number of columns in the flexibility matrixthat is equal to the total degrees of freedom of the structureand 119899119905119890 is the total number of truss elements

6 Shock and Vibration

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 5 Damage identification for 31-bar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 6 Damage identification for 31-bar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

Theoretically damage occurrence leads to increasing theSCM and consequently the index SCBFI119890 As a result inthis study by determining the parameter SCBFI119890 for eachelement of the structure an efficient indicator for estimatingthe presence and locating the damage in the element can bedefined Assuming that the set of the damage indices SCBFI

119890

(119890 = 1 2 119899119905119890) represents a sample population of a normallydistributed random variable a normalized damage index canbe defined as follows

SCBFI119899119890= [

(SCBFI119890minusmean (SCBFI))

std (SCBFI)]

119890 = 1 2 119899119905119890

(15)

where mean(SCBFI) and std(SCBFI) represent the meanand standard deviation of the vector of damage indicesrespectively

In order to obtain a more accurate damage extent for anelement the damage indicator of (15) needs to be furtherscaled as

SCBFI119899119890=

SCBFI119899119890

3 SCBFI119899 119890 = 1 2 119899119905119890 (16)

where sdot symbolizes the magnitude of a vectorThe process of constructing the SCBFI index can also be

briefly shown in Figure 1

Shock and Vibration 7

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 7 Damage identification for 31-bar truss for case 5 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

(1)

(3)

(5) (6)

(7) (8)

(9) (10)

(11) (12)

(13) (14)(15) (16)

(17) (18) (19) (20) (21) (22)

(2)

(4)

1

2

3

4

5

6

7

8

9

1011 12

13 1415 1617 18

19 2021 22

29

23 2425 2627

28

3031 32

33

34 35

36

37

38

39 40

41

42

43

44 45

46

47

600 in

540 in

480 in

420 in

360 in

240 in

120 in

0

90 in150 in

30 in

Y

X

60 in 60 in

Figure 8 Forty-seven-bar planar power line tower

8 Shock and Vibration

00102030405

1 5 9 13 17 21 25 29 33 37 41 45

Dam

age r

atio

Element number

(a)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0

02

04

06

(b)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(d)

Figure 9 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 10 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

4 Test Examples

In order to show the capabilities of the proposed method foridentifying the multiple-structural damage two illustrativetest examples are considered The first example is a 31-barplanar truss and the second one is a 47-bar planar trussThe effect of measurement noise on the performance of themethod is considered in the first example

41 Thirty-One-Bar Planar Truss The 31-bar planar trussshown in Figure 2 selected from [23] is modeled using theconventional finite element method without internal nodes

leading to 25 degrees of freedom The material density andelasticity modulus are 2770 kgm3 and 70GPa respectivelyDamage in the structure is simulated as a relative reductionin the elasticity modulus of individual bars Five differentdamage cases given in Table 1 are induced in the structure andthe proposed method is tested for each case Figures 3 4 5 6and 7 show the SCBFI value with respect to element numberfor damage cases 1 to 5 when one to four mode shapes areconsidered respectively

It can be observed that the new index achieves the actualsite of damage truthfully in most cases It is also revealed thatthe general configuration of identification charts does not

Shock and Vibration 9

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

0

02

04

06

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

001020304

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

001020304

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 11 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

00020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 12 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

0005

01015

02025

03

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

001020304

(b)

Figure 13 The SCBFI for 31-bar truss considering 1 noise for (a) case 1 and (b) case 2

10 Shock and Vibration

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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Shock and Vibration

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2 Shock and Vibration

damage has been introduced by Li et al [11] Compared withthe original flexibility matrix based approach the effect oftruncating higher-order modes can be considerably reducedin the new approach A method based on best achievableflexibility change with capability in detecting the locationand extent of structural damage has also been presentedby Yang and Sun [12] A method for structural damageidentification based on flexibility disassembly has also beenproposed by Yang [13]The efficiency of the proposedmethodhas been demonstrated by numerical examples A flexibilitybased damage index has been proposed for structural damagedetection by Zhang et al [14] Example of a 12-storey buildingmodel illustrated the efficiency of the proposed index forsuccessfully detecting damage locations in both the singleand multiple damage cases The flexibility matrix and strainenergy concepts of a structure have been used by Nobahariand Seyedpoor [15] in order to introduce a damage indicatorfor locating structural damage

In this paper a new index for structural damage detectionis proposed The principle of the index is based on compar-ison of structural elementrsquos strains obtained from two setsrelated to intact anddamaged structureThe calculation of thestrain is based on a flexibility matrix estimated from modalanalysis information Taking advantage of highly convergedflexibility matrix using only few vibration modes related tolow frequencies in the first phase and then determining theelemental strain using the flexibility coefficients develop arobust tool for damage localization of truss structures

2 Structural Damage Detection

In recent years many damage indices have been proposed toidentify structural damage In this paper a number of widelyused indices are first described and then the new proposeddamage index is introduced

21 Modal Assurance Criterion (MAC) and Coordinate ModalAssurance Criterion (COMAC) Based on the basic modalparameters of structures such as natural frequencies damp-ing ratios and mode shapes some coefficients derived fromthese parameters can be useful for damage detection TheMAC and COMAC factors [16] may be mentioned in thiscategory The factors are derived from mode shapes andexpress the correlation between two mode shapes obtainedfrom two sets Sufficient number of degrees of freedom(number of measurement points) are needed here to attaingood accuracy

Let [120593119860] and [120593

119861] be the first and second sets of measured

mode shapes in a matrix form of size 119899 times 119898119860and 119899 times 119898

119861

respectively 119898119860and 119898

119861are the numbers of mode shapes

considered in the respective sets and 119899 is the number ofmeasurement points The MAC factor is then defined for themode shapes 119895 and 119896 as follows

MAC(119895119896)

=

100381610038161003816100381610038161003816sum119899

119894=1[120593119860]119895

119894sdot [120593119861]119896

119894

100381610038161003816100381610038161003816

2

sum119899

119894=1([120593119860]119895

119894)2

sdot sum119899

119894=1([120593119861]119896

119894)2 (1)

where 119895 = 1 119898119860

and 119896 = 1 119898119861 [120593119860]119895

119894and

[120593119861]119896

119894are the 119894th components of the modes [120593

119860]119895 and [120593

119861]119896

respectively The MAC(119895119896)

factor indicates the degree ofcorrelation between the 119895th mode of the first set 119860 and the119896th mode of the second set 119861 The MAC values vary from0 to 1 with 0 for no correlation and 1 for full correlationTherefore if the eigenvectors [120593

119860] and [120593

119861] are identical the

corresponding MAC values will be close to 1 thus indicatingthe full correlation between the two modes The deviation ofthe factors from 1 can be interpreted as a damage indicator ina structure

The COMAC factors are generally used to identify wherethe mode shapes of the structure from two sets of mea-surements do not correlate If the modal displacements ina coordinate 119895 from two sets of measurements are identicalthe COMAC factor is close to 1 for this coordinate A largedeviation from unity can be interpreted as damage indicationin the structure For the coordinate 119901 of a structure using 119898

mode shapes the COMAC factor is defined as follows

COMAC119901=

10038161003816100381610038161003816sum119898

119894=1[120593119860]119901

119894sdot [120593119861]119901

119894

10038161003816100381610038161003816

2

sum119898

119894=1([120593119860]119901

119894)2

sdot sum119898

119894=1([120593119861]119901

119894)2 (2)

In practice only a few mode shapes of a structure canbe measured As a result a method capable of predictingstructural damage that requires a limited number of modeshapes would be more efficient

22 Flexibility Method It has been proved that the presenceof damage in a structure increases its flexibility So anychange observed in the flexibility matrix can be interpretedas a damage indication in the structure and allows one toidentify damage [17] Therefore another class of damageidentificationmethods is based onusing the flexibilitymatrixThe flexibility matrix 119865 is the inverse of the stiffness matrix119870 relating the applied static forces 119891 to resulting structuraldisplacements 119906 as

119906 = [119865] 119891 (3)

The flexibility matrix can be also dynamically measured frommodal analysis data The relationship between the flexibilitymatrix 119865 and the dynamic characteristics of a structure canbe given by [15 17]

119865 = [120593] [Λ]minus1[120593]119879≃

119899119898

sum119894=1

1

1205962119894

120593119894120593119879

119894 (4)

where [120593] is the mode shape matrix [Λ] = diag(11205962) isa diagonal matrix 120596

119894is the 119894th circular frequency 120593

119894is the

mass-normalized 119894th mode shape of the structure and nm isthe number of measured modes

From (4) one can see that the modal contribution to theflexibility matrix decreases as the frequency increases On theother hand the flexibility matrix converges rapidly with theincrease of the values of the frequenciesTherefore from onlya few of lowermodes a good estimate for the flexibilitymatrixcan be achieved

Shock and Vibration 3

The principle of flexibility method is based on a compar-ison of the flexibility matrices from two sets of mode shapesIf 119865ℎand 119865

119889are the flexibility matrices corresponding to

the healthy and damaged states of the structure a flexibilitychange matrix Δ119865 can be defined as the difference of the twomatrices

Δ119865 = 119865119889minus 119865ℎ (5)

For each degree of freedom let 120575119895be the maximum absolute

value of the elements in the corresponding column of Δ119865

120575119895= max 10038161003816100381610038161003816120575119894119895

10038161003816100381610038161003816 (6)

where 120575119894119895are the elements of Δ119865 In order to identify damage

in the structure the quantity 120575119895can be used as a damage

indicator [17]

23 Modal Strain Energy Based Method The methods basedon modal strain energy of a structure have been commonlyused in damage detection [17ndash19] Since the mode shapevectors are equivalent to nodal displacements of a vibratingstructure therefore in each element of the structure strainenergy is stored The strain energy of a structure due tomode shape vectors is usually referred to as modal strainenergy (MSE) and can be considered as a valuable parameterfor damage identification The modal strain energy of ethelement in 119894th mode of the structure can be expressed as [19]

mse119890119894=

1

2120593119890119879

119894119870119890120593119890

119894119894 = 1 2 119899119898 119890 = 1 2 119899119905119890 (7)

where119870119890 is the stiffnessmatrix of 119890th element of the structureand 120593

119890

119894is the vector of corresponding nodal displacements of

element 119890 in mode 119894A normalized form of MSE considering 119899119898 vibrating

modes of the structure proposed in [19] can be given as

mnmse119890 =sum119899119898

119894=1(mse119890119894sum119899119905119890

119890=1mse119890119894)

119899119898 119890 = 1 119899119905119890 (8)

The damage occurrence increases the MSE and consequentlythe efficient parameter mnmse119890 So by determining theparameter mnmse119890 for each element of healthy and damagedstructures an efficient indicator for identifying the damage inthe element can be defined [19]

3 The Proposed Strain Change Based onFlexibility Index (SCBFI)

In this study a new damage detection index based on consid-ering strain changes in a structural element due to damageis developed The new index SCBFI proposed is capable ofidentifying and locating the multiple-structural damage intruss structures The principle of the new index is based onevaluating the changes of strain in structural elements butthe method of computing the strain is completely differentfrom the usual methodsThe nodal displacement vector usedfor computing the strain obtains from the flexibility matrix of

the structure Moreover the flexibility matrix is determinedfrommodal analysis information including mode shapes andnatural frequencies Taking advantage of rapid convergenceof the flexibility matrix in terms of the number of vibrationmodes helps the proposed index to identify the structuraldamage with more efficiency and lower computational cost

As the first step for constructing the proposed damageindex a modal analysis is required to be performed Themodal analysis is a tool to determine the natural frequenciesand mode shapes of a structure [20] It has the mathematicalform of

(119870 minus 1205962

119894119872)120593119894= 0 119894 = 1 2 119899119898 (9)

where 119870 and 119872 are the stiffness and mass matrices of thestructure respectively Also 120596

119894and 120593

119894are the 119894th circular fre-

quency and mode shape vector of the structure respectivelyAt the second step the flexibility matrix of healthy and

damaged structure (FMH FMD) related to the dynamiccharacteristics of the healthy and damaged structure can beapproximated as

FMH ≃

119899119898

sum119894=1

1

(120596ℎ119894)2120593ℎ

119894120593ℎ

119894

119879

FMD ≃

119899119898

sum119894=1

1

(120596119889119894)2120593119889

119894120593119889

119894

119879

(10)

where 120596ℎ119894and 120596

119889

119894are the 119894th circular frequency of healthy and

damaged structure respectively 120593ℎ119894and 120593

119889

119894are the 119894th mode

shape vector of healthy and damaged structure respectivelyand 119899119898 is the number of vibrating modes considered

It can be observed that all components of the modeshapes are required to be measured and it is not a realisticassumption for operational damage detection However foractual use of the suggested method it is not needed tomeasure the full set of mode shapes The mode shapes ofthe damaged structure in partial degrees of freedom arefirst measured and then the incomplete mode shapes areexpanded tomatch all degrees of freedomof the structure by acommon technique such as a dynamic condensation method[21]

Since each column of the flexibility matrix representsthe displacement pattern of the structure associated with aunit force applied at the corresponding DOF of that columntherefore they can be used as nodal displacements to calculatethe strains of structural elements The strain of each elementof a 2-D truss structure can be expressed as [22]

120598 =1

119871[minus cos 120579 minus sin 120579 cos 120579 sin 120579 ]

1198891119909

1198891119910

1198892119909

1198892119910

(11)

where 120598 is the strain of truss element 119871 is the length ofelement 120579 is the angle between local and global coordinatesand 119889

119879= 1198891119909 1198891119910 1198892119909 1198892119910 is the nodal displacement vector

of the element

4 Shock and Vibration

Start

Modal analysis (natural frequencies and mode shapes of

damaged structure)

Calculate the flexibility matrix ofdamaged structure (FMD) using

a few mode shapes

Consider each column of FMD as displacement pattern for damaged structure

Arrange the strain matrix of damaged structure (SMD)

Determine the strain change matrix as

Modal analysis (natural frequencies and mode shapes of

healthy structure)

Calculate the flexibility matrix of healthy structure (FMH) using a

few mode shapes

Consider each column of FMH as displacement pattern for healthy structure

Arrange the strain matrix of healthy structure (SMH)

SCM = SMD minus SMH

Calculate the absolute mean value of each rowof matrix SCM as

SCBFIe = | |sumnci=1 SCM i

e

nc| e =

using (15) and (16)Normalize and scale the SCBFIe

1 2 nte

Figure 1 The process for constructing the SCBFI index

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

152

m

152m

Figure 2 Planar truss having 31 elements

Shock and Vibration 5

001020304

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

SCBF

I

Element number

(a)

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(c)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(d)

Figure 3 Damage identification for 31-bar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 4 Damage identification for 31-bar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

So at the next step the strain of eth element for 119894thcolumn of healthy and damaged structure based on thecolumnar coefficients of the flexibility matrix can be deter-mined as

SMH119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890ℎ

SMD119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890119889

(12)

where SMH119894119890is the strain of 119890th element related to 119894th column

of FMH SMD119894119890is the strain of 119890th element related to 119894th

column of FMD 119889119894119890ℎ and 119889

119894

119890119889 are the nodal displacement

vectors of 119890th element associatedwith the 119894th column of flexi-bility matrix for healthy and damaged structure respectively120579119890is the angle between local and global coordinates of 119890th

element and 119871119890is the length of 119890th element

Now the strain change matrix SCM can be defined as thedifference between the strain matrix of damaged structureSMD and the strain matrix of healthy structure SMH as

SCM = SMD minus SMH (13)

The 119894th rowof SCMmatrix represents the strain changes of 119894thelement of structure for unit loads applied at different DOFsof the structure The SCBFI index can now be defined as acolumnar vector containing the absolute mean value of eachrow of SCMmatrix given by

SCBFI119890=

100381610038161003816100381610038161003816100381610038161003816

sum119899119888

119894=1SCM119894119890

119899119888

100381610038161003816100381610038161003816100381610038161003816 119890 = 1 2 119899119905119890 (14)

where 119899119888 is the number of columns in the flexibility matrixthat is equal to the total degrees of freedom of the structureand 119899119905119890 is the total number of truss elements

6 Shock and Vibration

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 5 Damage identification for 31-bar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 6 Damage identification for 31-bar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

Theoretically damage occurrence leads to increasing theSCM and consequently the index SCBFI119890 As a result inthis study by determining the parameter SCBFI119890 for eachelement of the structure an efficient indicator for estimatingthe presence and locating the damage in the element can bedefined Assuming that the set of the damage indices SCBFI

119890

(119890 = 1 2 119899119905119890) represents a sample population of a normallydistributed random variable a normalized damage index canbe defined as follows

SCBFI119899119890= [

(SCBFI119890minusmean (SCBFI))

std (SCBFI)]

119890 = 1 2 119899119905119890

(15)

where mean(SCBFI) and std(SCBFI) represent the meanand standard deviation of the vector of damage indicesrespectively

In order to obtain a more accurate damage extent for anelement the damage indicator of (15) needs to be furtherscaled as

SCBFI119899119890=

SCBFI119899119890

3 SCBFI119899 119890 = 1 2 119899119905119890 (16)

where sdot symbolizes the magnitude of a vectorThe process of constructing the SCBFI index can also be

briefly shown in Figure 1

Shock and Vibration 7

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 7 Damage identification for 31-bar truss for case 5 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

(1)

(3)

(5) (6)

(7) (8)

(9) (10)

(11) (12)

(13) (14)(15) (16)

(17) (18) (19) (20) (21) (22)

(2)

(4)

1

2

3

4

5

6

7

8

9

1011 12

13 1415 1617 18

19 2021 22

29

23 2425 2627

28

3031 32

33

34 35

36

37

38

39 40

41

42

43

44 45

46

47

600 in

540 in

480 in

420 in

360 in

240 in

120 in

0

90 in150 in

30 in

Y

X

60 in 60 in

Figure 8 Forty-seven-bar planar power line tower

8 Shock and Vibration

00102030405

1 5 9 13 17 21 25 29 33 37 41 45

Dam

age r

atio

Element number

(a)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0

02

04

06

(b)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(d)

Figure 9 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 10 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

4 Test Examples

In order to show the capabilities of the proposed method foridentifying the multiple-structural damage two illustrativetest examples are considered The first example is a 31-barplanar truss and the second one is a 47-bar planar trussThe effect of measurement noise on the performance of themethod is considered in the first example

41 Thirty-One-Bar Planar Truss The 31-bar planar trussshown in Figure 2 selected from [23] is modeled using theconventional finite element method without internal nodes

leading to 25 degrees of freedom The material density andelasticity modulus are 2770 kgm3 and 70GPa respectivelyDamage in the structure is simulated as a relative reductionin the elasticity modulus of individual bars Five differentdamage cases given in Table 1 are induced in the structure andthe proposed method is tested for each case Figures 3 4 5 6and 7 show the SCBFI value with respect to element numberfor damage cases 1 to 5 when one to four mode shapes areconsidered respectively

It can be observed that the new index achieves the actualsite of damage truthfully in most cases It is also revealed thatthe general configuration of identification charts does not

Shock and Vibration 9

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

0

02

04

06

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

001020304

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

001020304

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 11 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

00020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 12 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

0005

01015

02025

03

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

001020304

(b)

Figure 13 The SCBFI for 31-bar truss considering 1 noise for (a) case 1 and (b) case 2

10 Shock and Vibration

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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Shock and Vibration

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DistributedSensor Networks

International Journal of

Shock and Vibration 3

The principle of flexibility method is based on a compar-ison of the flexibility matrices from two sets of mode shapesIf 119865ℎand 119865

119889are the flexibility matrices corresponding to

the healthy and damaged states of the structure a flexibilitychange matrix Δ119865 can be defined as the difference of the twomatrices

Δ119865 = 119865119889minus 119865ℎ (5)

For each degree of freedom let 120575119895be the maximum absolute

value of the elements in the corresponding column of Δ119865

120575119895= max 10038161003816100381610038161003816120575119894119895

10038161003816100381610038161003816 (6)

where 120575119894119895are the elements of Δ119865 In order to identify damage

in the structure the quantity 120575119895can be used as a damage

indicator [17]

23 Modal Strain Energy Based Method The methods basedon modal strain energy of a structure have been commonlyused in damage detection [17ndash19] Since the mode shapevectors are equivalent to nodal displacements of a vibratingstructure therefore in each element of the structure strainenergy is stored The strain energy of a structure due tomode shape vectors is usually referred to as modal strainenergy (MSE) and can be considered as a valuable parameterfor damage identification The modal strain energy of ethelement in 119894th mode of the structure can be expressed as [19]

mse119890119894=

1

2120593119890119879

119894119870119890120593119890

119894119894 = 1 2 119899119898 119890 = 1 2 119899119905119890 (7)

where119870119890 is the stiffnessmatrix of 119890th element of the structureand 120593

119890

119894is the vector of corresponding nodal displacements of

element 119890 in mode 119894A normalized form of MSE considering 119899119898 vibrating

modes of the structure proposed in [19] can be given as

mnmse119890 =sum119899119898

119894=1(mse119890119894sum119899119905119890

119890=1mse119890119894)

119899119898 119890 = 1 119899119905119890 (8)

The damage occurrence increases the MSE and consequentlythe efficient parameter mnmse119890 So by determining theparameter mnmse119890 for each element of healthy and damagedstructures an efficient indicator for identifying the damage inthe element can be defined [19]

3 The Proposed Strain Change Based onFlexibility Index (SCBFI)

In this study a new damage detection index based on consid-ering strain changes in a structural element due to damageis developed The new index SCBFI proposed is capable ofidentifying and locating the multiple-structural damage intruss structures The principle of the new index is based onevaluating the changes of strain in structural elements butthe method of computing the strain is completely differentfrom the usual methodsThe nodal displacement vector usedfor computing the strain obtains from the flexibility matrix of

the structure Moreover the flexibility matrix is determinedfrommodal analysis information including mode shapes andnatural frequencies Taking advantage of rapid convergenceof the flexibility matrix in terms of the number of vibrationmodes helps the proposed index to identify the structuraldamage with more efficiency and lower computational cost

As the first step for constructing the proposed damageindex a modal analysis is required to be performed Themodal analysis is a tool to determine the natural frequenciesand mode shapes of a structure [20] It has the mathematicalform of

(119870 minus 1205962

119894119872)120593119894= 0 119894 = 1 2 119899119898 (9)

where 119870 and 119872 are the stiffness and mass matrices of thestructure respectively Also 120596

119894and 120593

119894are the 119894th circular fre-

quency and mode shape vector of the structure respectivelyAt the second step the flexibility matrix of healthy and

damaged structure (FMH FMD) related to the dynamiccharacteristics of the healthy and damaged structure can beapproximated as

FMH ≃

119899119898

sum119894=1

1

(120596ℎ119894)2120593ℎ

119894120593ℎ

119894

119879

FMD ≃

119899119898

sum119894=1

1

(120596119889119894)2120593119889

119894120593119889

119894

119879

(10)

where 120596ℎ119894and 120596

119889

119894are the 119894th circular frequency of healthy and

damaged structure respectively 120593ℎ119894and 120593

119889

119894are the 119894th mode

shape vector of healthy and damaged structure respectivelyand 119899119898 is the number of vibrating modes considered

It can be observed that all components of the modeshapes are required to be measured and it is not a realisticassumption for operational damage detection However foractual use of the suggested method it is not needed tomeasure the full set of mode shapes The mode shapes ofthe damaged structure in partial degrees of freedom arefirst measured and then the incomplete mode shapes areexpanded tomatch all degrees of freedomof the structure by acommon technique such as a dynamic condensation method[21]

Since each column of the flexibility matrix representsthe displacement pattern of the structure associated with aunit force applied at the corresponding DOF of that columntherefore they can be used as nodal displacements to calculatethe strains of structural elements The strain of each elementof a 2-D truss structure can be expressed as [22]

120598 =1

119871[minus cos 120579 minus sin 120579 cos 120579 sin 120579 ]

1198891119909

1198891119910

1198892119909

1198892119910

(11)

where 120598 is the strain of truss element 119871 is the length ofelement 120579 is the angle between local and global coordinatesand 119889

119879= 1198891119909 1198891119910 1198892119909 1198892119910 is the nodal displacement vector

of the element

4 Shock and Vibration

Start

Modal analysis (natural frequencies and mode shapes of

damaged structure)

Calculate the flexibility matrix ofdamaged structure (FMD) using

a few mode shapes

Consider each column of FMD as displacement pattern for damaged structure

Arrange the strain matrix of damaged structure (SMD)

Determine the strain change matrix as

Modal analysis (natural frequencies and mode shapes of

healthy structure)

Calculate the flexibility matrix of healthy structure (FMH) using a

few mode shapes

Consider each column of FMH as displacement pattern for healthy structure

Arrange the strain matrix of healthy structure (SMH)

SCM = SMD minus SMH

Calculate the absolute mean value of each rowof matrix SCM as

SCBFIe = | |sumnci=1 SCM i

e

nc| e =

using (15) and (16)Normalize and scale the SCBFIe

1 2 nte

Figure 1 The process for constructing the SCBFI index

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

152

m

152m

Figure 2 Planar truss having 31 elements

Shock and Vibration 5

001020304

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

SCBF

I

Element number

(a)

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(c)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(d)

Figure 3 Damage identification for 31-bar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 4 Damage identification for 31-bar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

So at the next step the strain of eth element for 119894thcolumn of healthy and damaged structure based on thecolumnar coefficients of the flexibility matrix can be deter-mined as

SMH119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890ℎ

SMD119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890119889

(12)

where SMH119894119890is the strain of 119890th element related to 119894th column

of FMH SMD119894119890is the strain of 119890th element related to 119894th

column of FMD 119889119894119890ℎ and 119889

119894

119890119889 are the nodal displacement

vectors of 119890th element associatedwith the 119894th column of flexi-bility matrix for healthy and damaged structure respectively120579119890is the angle between local and global coordinates of 119890th

element and 119871119890is the length of 119890th element

Now the strain change matrix SCM can be defined as thedifference between the strain matrix of damaged structureSMD and the strain matrix of healthy structure SMH as

SCM = SMD minus SMH (13)

The 119894th rowof SCMmatrix represents the strain changes of 119894thelement of structure for unit loads applied at different DOFsof the structure The SCBFI index can now be defined as acolumnar vector containing the absolute mean value of eachrow of SCMmatrix given by

SCBFI119890=

100381610038161003816100381610038161003816100381610038161003816

sum119899119888

119894=1SCM119894119890

119899119888

100381610038161003816100381610038161003816100381610038161003816 119890 = 1 2 119899119905119890 (14)

where 119899119888 is the number of columns in the flexibility matrixthat is equal to the total degrees of freedom of the structureand 119899119905119890 is the total number of truss elements

6 Shock and Vibration

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 5 Damage identification for 31-bar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 6 Damage identification for 31-bar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

Theoretically damage occurrence leads to increasing theSCM and consequently the index SCBFI119890 As a result inthis study by determining the parameter SCBFI119890 for eachelement of the structure an efficient indicator for estimatingthe presence and locating the damage in the element can bedefined Assuming that the set of the damage indices SCBFI

119890

(119890 = 1 2 119899119905119890) represents a sample population of a normallydistributed random variable a normalized damage index canbe defined as follows

SCBFI119899119890= [

(SCBFI119890minusmean (SCBFI))

std (SCBFI)]

119890 = 1 2 119899119905119890

(15)

where mean(SCBFI) and std(SCBFI) represent the meanand standard deviation of the vector of damage indicesrespectively

In order to obtain a more accurate damage extent for anelement the damage indicator of (15) needs to be furtherscaled as

SCBFI119899119890=

SCBFI119899119890

3 SCBFI119899 119890 = 1 2 119899119905119890 (16)

where sdot symbolizes the magnitude of a vectorThe process of constructing the SCBFI index can also be

briefly shown in Figure 1

Shock and Vibration 7

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 7 Damage identification for 31-bar truss for case 5 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

(1)

(3)

(5) (6)

(7) (8)

(9) (10)

(11) (12)

(13) (14)(15) (16)

(17) (18) (19) (20) (21) (22)

(2)

(4)

1

2

3

4

5

6

7

8

9

1011 12

13 1415 1617 18

19 2021 22

29

23 2425 2627

28

3031 32

33

34 35

36

37

38

39 40

41

42

43

44 45

46

47

600 in

540 in

480 in

420 in

360 in

240 in

120 in

0

90 in150 in

30 in

Y

X

60 in 60 in

Figure 8 Forty-seven-bar planar power line tower

8 Shock and Vibration

00102030405

1 5 9 13 17 21 25 29 33 37 41 45

Dam

age r

atio

Element number

(a)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0

02

04

06

(b)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(d)

Figure 9 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 10 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

4 Test Examples

In order to show the capabilities of the proposed method foridentifying the multiple-structural damage two illustrativetest examples are considered The first example is a 31-barplanar truss and the second one is a 47-bar planar trussThe effect of measurement noise on the performance of themethod is considered in the first example

41 Thirty-One-Bar Planar Truss The 31-bar planar trussshown in Figure 2 selected from [23] is modeled using theconventional finite element method without internal nodes

leading to 25 degrees of freedom The material density andelasticity modulus are 2770 kgm3 and 70GPa respectivelyDamage in the structure is simulated as a relative reductionin the elasticity modulus of individual bars Five differentdamage cases given in Table 1 are induced in the structure andthe proposed method is tested for each case Figures 3 4 5 6and 7 show the SCBFI value with respect to element numberfor damage cases 1 to 5 when one to four mode shapes areconsidered respectively

It can be observed that the new index achieves the actualsite of damage truthfully in most cases It is also revealed thatthe general configuration of identification charts does not

Shock and Vibration 9

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

0

02

04

06

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

001020304

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

001020304

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 11 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

00020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 12 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

0005

01015

02025

03

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

001020304

(b)

Figure 13 The SCBFI for 31-bar truss considering 1 noise for (a) case 1 and (b) case 2

10 Shock and Vibration

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

Start

Modal analysis (natural frequencies and mode shapes of

damaged structure)

Calculate the flexibility matrix ofdamaged structure (FMD) using

a few mode shapes

Consider each column of FMD as displacement pattern for damaged structure

Arrange the strain matrix of damaged structure (SMD)

Determine the strain change matrix as

Modal analysis (natural frequencies and mode shapes of

healthy structure)

Calculate the flexibility matrix of healthy structure (FMH) using a

few mode shapes

Consider each column of FMH as displacement pattern for healthy structure

Arrange the strain matrix of healthy structure (SMH)

SCM = SMD minus SMH

Calculate the absolute mean value of each rowof matrix SCM as

SCBFIe = | |sumnci=1 SCM i

e

nc| e =

using (15) and (16)Normalize and scale the SCBFIe

1 2 nte

Figure 1 The process for constructing the SCBFI index

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

152

m

152m

Figure 2 Planar truss having 31 elements

Shock and Vibration 5

001020304

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

SCBF

I

Element number

(a)

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(c)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(d)

Figure 3 Damage identification for 31-bar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 4 Damage identification for 31-bar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

So at the next step the strain of eth element for 119894thcolumn of healthy and damaged structure based on thecolumnar coefficients of the flexibility matrix can be deter-mined as

SMH119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890ℎ

SMD119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890119889

(12)

where SMH119894119890is the strain of 119890th element related to 119894th column

of FMH SMD119894119890is the strain of 119890th element related to 119894th

column of FMD 119889119894119890ℎ and 119889

119894

119890119889 are the nodal displacement

vectors of 119890th element associatedwith the 119894th column of flexi-bility matrix for healthy and damaged structure respectively120579119890is the angle between local and global coordinates of 119890th

element and 119871119890is the length of 119890th element

Now the strain change matrix SCM can be defined as thedifference between the strain matrix of damaged structureSMD and the strain matrix of healthy structure SMH as

SCM = SMD minus SMH (13)

The 119894th rowof SCMmatrix represents the strain changes of 119894thelement of structure for unit loads applied at different DOFsof the structure The SCBFI index can now be defined as acolumnar vector containing the absolute mean value of eachrow of SCMmatrix given by

SCBFI119890=

100381610038161003816100381610038161003816100381610038161003816

sum119899119888

119894=1SCM119894119890

119899119888

100381610038161003816100381610038161003816100381610038161003816 119890 = 1 2 119899119905119890 (14)

where 119899119888 is the number of columns in the flexibility matrixthat is equal to the total degrees of freedom of the structureand 119899119905119890 is the total number of truss elements

6 Shock and Vibration

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 5 Damage identification for 31-bar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 6 Damage identification for 31-bar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

Theoretically damage occurrence leads to increasing theSCM and consequently the index SCBFI119890 As a result inthis study by determining the parameter SCBFI119890 for eachelement of the structure an efficient indicator for estimatingthe presence and locating the damage in the element can bedefined Assuming that the set of the damage indices SCBFI

119890

(119890 = 1 2 119899119905119890) represents a sample population of a normallydistributed random variable a normalized damage index canbe defined as follows

SCBFI119899119890= [

(SCBFI119890minusmean (SCBFI))

std (SCBFI)]

119890 = 1 2 119899119905119890

(15)

where mean(SCBFI) and std(SCBFI) represent the meanand standard deviation of the vector of damage indicesrespectively

In order to obtain a more accurate damage extent for anelement the damage indicator of (15) needs to be furtherscaled as

SCBFI119899119890=

SCBFI119899119890

3 SCBFI119899 119890 = 1 2 119899119905119890 (16)

where sdot symbolizes the magnitude of a vectorThe process of constructing the SCBFI index can also be

briefly shown in Figure 1

Shock and Vibration 7

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 7 Damage identification for 31-bar truss for case 5 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

(1)

(3)

(5) (6)

(7) (8)

(9) (10)

(11) (12)

(13) (14)(15) (16)

(17) (18) (19) (20) (21) (22)

(2)

(4)

1

2

3

4

5

6

7

8

9

1011 12

13 1415 1617 18

19 2021 22

29

23 2425 2627

28

3031 32

33

34 35

36

37

38

39 40

41

42

43

44 45

46

47

600 in

540 in

480 in

420 in

360 in

240 in

120 in

0

90 in150 in

30 in

Y

X

60 in 60 in

Figure 8 Forty-seven-bar planar power line tower

8 Shock and Vibration

00102030405

1 5 9 13 17 21 25 29 33 37 41 45

Dam

age r

atio

Element number

(a)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0

02

04

06

(b)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(d)

Figure 9 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 10 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

4 Test Examples

In order to show the capabilities of the proposed method foridentifying the multiple-structural damage two illustrativetest examples are considered The first example is a 31-barplanar truss and the second one is a 47-bar planar trussThe effect of measurement noise on the performance of themethod is considered in the first example

41 Thirty-One-Bar Planar Truss The 31-bar planar trussshown in Figure 2 selected from [23] is modeled using theconventional finite element method without internal nodes

leading to 25 degrees of freedom The material density andelasticity modulus are 2770 kgm3 and 70GPa respectivelyDamage in the structure is simulated as a relative reductionin the elasticity modulus of individual bars Five differentdamage cases given in Table 1 are induced in the structure andthe proposed method is tested for each case Figures 3 4 5 6and 7 show the SCBFI value with respect to element numberfor damage cases 1 to 5 when one to four mode shapes areconsidered respectively

It can be observed that the new index achieves the actualsite of damage truthfully in most cases It is also revealed thatthe general configuration of identification charts does not

Shock and Vibration 9

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

0

02

04

06

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

001020304

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

001020304

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 11 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

00020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 12 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

0005

01015

02025

03

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

001020304

(b)

Figure 13 The SCBFI for 31-bar truss considering 1 noise for (a) case 1 and (b) case 2

10 Shock and Vibration

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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DistributedSensor Networks

International Journal of

Shock and Vibration 5

001020304

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

SCBF

I

Element number

(a)

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(c)

Identified damageInduced damage

0010203

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(d)

Figure 3 Damage identification for 31-bar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 4 Damage identification for 31-bar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

So at the next step the strain of eth element for 119894thcolumn of healthy and damaged structure based on thecolumnar coefficients of the flexibility matrix can be deter-mined as

SMH119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890ℎ

SMD119894119890=

1

119871119890

[minus cos 120579119890

minus sin 120579119890

cos 120579119890sin 120579119890 ] 119889119894

119890119889

(12)

where SMH119894119890is the strain of 119890th element related to 119894th column

of FMH SMD119894119890is the strain of 119890th element related to 119894th

column of FMD 119889119894119890ℎ and 119889

119894

119890119889 are the nodal displacement

vectors of 119890th element associatedwith the 119894th column of flexi-bility matrix for healthy and damaged structure respectively120579119890is the angle between local and global coordinates of 119890th

element and 119871119890is the length of 119890th element

Now the strain change matrix SCM can be defined as thedifference between the strain matrix of damaged structureSMD and the strain matrix of healthy structure SMH as

SCM = SMD minus SMH (13)

The 119894th rowof SCMmatrix represents the strain changes of 119894thelement of structure for unit loads applied at different DOFsof the structure The SCBFI index can now be defined as acolumnar vector containing the absolute mean value of eachrow of SCMmatrix given by

SCBFI119890=

100381610038161003816100381610038161003816100381610038161003816

sum119899119888

119894=1SCM119894119890

119899119888

100381610038161003816100381610038161003816100381610038161003816 119890 = 1 2 119899119905119890 (14)

where 119899119888 is the number of columns in the flexibility matrixthat is equal to the total degrees of freedom of the structureand 119899119905119890 is the total number of truss elements

6 Shock and Vibration

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 5 Damage identification for 31-bar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 6 Damage identification for 31-bar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

Theoretically damage occurrence leads to increasing theSCM and consequently the index SCBFI119890 As a result inthis study by determining the parameter SCBFI119890 for eachelement of the structure an efficient indicator for estimatingthe presence and locating the damage in the element can bedefined Assuming that the set of the damage indices SCBFI

119890

(119890 = 1 2 119899119905119890) represents a sample population of a normallydistributed random variable a normalized damage index canbe defined as follows

SCBFI119899119890= [

(SCBFI119890minusmean (SCBFI))

std (SCBFI)]

119890 = 1 2 119899119905119890

(15)

where mean(SCBFI) and std(SCBFI) represent the meanand standard deviation of the vector of damage indicesrespectively

In order to obtain a more accurate damage extent for anelement the damage indicator of (15) needs to be furtherscaled as

SCBFI119899119890=

SCBFI119899119890

3 SCBFI119899 119890 = 1 2 119899119905119890 (16)

where sdot symbolizes the magnitude of a vectorThe process of constructing the SCBFI index can also be

briefly shown in Figure 1

Shock and Vibration 7

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 7 Damage identification for 31-bar truss for case 5 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

(1)

(3)

(5) (6)

(7) (8)

(9) (10)

(11) (12)

(13) (14)(15) (16)

(17) (18) (19) (20) (21) (22)

(2)

(4)

1

2

3

4

5

6

7

8

9

1011 12

13 1415 1617 18

19 2021 22

29

23 2425 2627

28

3031 32

33

34 35

36

37

38

39 40

41

42

43

44 45

46

47

600 in

540 in

480 in

420 in

360 in

240 in

120 in

0

90 in150 in

30 in

Y

X

60 in 60 in

Figure 8 Forty-seven-bar planar power line tower

8 Shock and Vibration

00102030405

1 5 9 13 17 21 25 29 33 37 41 45

Dam

age r

atio

Element number

(a)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0

02

04

06

(b)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(d)

Figure 9 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 10 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

4 Test Examples

In order to show the capabilities of the proposed method foridentifying the multiple-structural damage two illustrativetest examples are considered The first example is a 31-barplanar truss and the second one is a 47-bar planar trussThe effect of measurement noise on the performance of themethod is considered in the first example

41 Thirty-One-Bar Planar Truss The 31-bar planar trussshown in Figure 2 selected from [23] is modeled using theconventional finite element method without internal nodes

leading to 25 degrees of freedom The material density andelasticity modulus are 2770 kgm3 and 70GPa respectivelyDamage in the structure is simulated as a relative reductionin the elasticity modulus of individual bars Five differentdamage cases given in Table 1 are induced in the structure andthe proposed method is tested for each case Figures 3 4 5 6and 7 show the SCBFI value with respect to element numberfor damage cases 1 to 5 when one to four mode shapes areconsidered respectively

It can be observed that the new index achieves the actualsite of damage truthfully in most cases It is also revealed thatthe general configuration of identification charts does not

Shock and Vibration 9

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

0

02

04

06

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

001020304

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

001020304

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 11 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

00020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 12 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

0005

01015

02025

03

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

001020304

(b)

Figure 13 The SCBFI for 31-bar truss considering 1 noise for (a) case 1 and (b) case 2

10 Shock and Vibration

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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

6 Shock and Vibration

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 5 Damage identification for 31-bar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 6 Damage identification for 31-bar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

Theoretically damage occurrence leads to increasing theSCM and consequently the index SCBFI119890 As a result inthis study by determining the parameter SCBFI119890 for eachelement of the structure an efficient indicator for estimatingthe presence and locating the damage in the element can bedefined Assuming that the set of the damage indices SCBFI

119890

(119890 = 1 2 119899119905119890) represents a sample population of a normallydistributed random variable a normalized damage index canbe defined as follows

SCBFI119899119890= [

(SCBFI119890minusmean (SCBFI))

std (SCBFI)]

119890 = 1 2 119899119905119890

(15)

where mean(SCBFI) and std(SCBFI) represent the meanand standard deviation of the vector of damage indicesrespectively

In order to obtain a more accurate damage extent for anelement the damage indicator of (15) needs to be furtherscaled as

SCBFI119899119890=

SCBFI119899119890

3 SCBFI119899 119890 = 1 2 119899119905119890 (16)

where sdot symbolizes the magnitude of a vectorThe process of constructing the SCBFI index can also be

briefly shown in Figure 1

Shock and Vibration 7

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 7 Damage identification for 31-bar truss for case 5 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

(1)

(3)

(5) (6)

(7) (8)

(9) (10)

(11) (12)

(13) (14)(15) (16)

(17) (18) (19) (20) (21) (22)

(2)

(4)

1

2

3

4

5

6

7

8

9

1011 12

13 1415 1617 18

19 2021 22

29

23 2425 2627

28

3031 32

33

34 35

36

37

38

39 40

41

42

43

44 45

46

47

600 in

540 in

480 in

420 in

360 in

240 in

120 in

0

90 in150 in

30 in

Y

X

60 in 60 in

Figure 8 Forty-seven-bar planar power line tower

8 Shock and Vibration

00102030405

1 5 9 13 17 21 25 29 33 37 41 45

Dam

age r

atio

Element number

(a)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0

02

04

06

(b)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(d)

Figure 9 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 10 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

4 Test Examples

In order to show the capabilities of the proposed method foridentifying the multiple-structural damage two illustrativetest examples are considered The first example is a 31-barplanar truss and the second one is a 47-bar planar trussThe effect of measurement noise on the performance of themethod is considered in the first example

41 Thirty-One-Bar Planar Truss The 31-bar planar trussshown in Figure 2 selected from [23] is modeled using theconventional finite element method without internal nodes

leading to 25 degrees of freedom The material density andelasticity modulus are 2770 kgm3 and 70GPa respectivelyDamage in the structure is simulated as a relative reductionin the elasticity modulus of individual bars Five differentdamage cases given in Table 1 are induced in the structure andthe proposed method is tested for each case Figures 3 4 5 6and 7 show the SCBFI value with respect to element numberfor damage cases 1 to 5 when one to four mode shapes areconsidered respectively

It can be observed that the new index achieves the actualsite of damage truthfully in most cases It is also revealed thatthe general configuration of identification charts does not

Shock and Vibration 9

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

0

02

04

06

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

001020304

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

001020304

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 11 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

00020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 12 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

0005

01015

02025

03

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

001020304

(b)

Figure 13 The SCBFI for 31-bar truss considering 1 noise for (a) case 1 and (b) case 2

10 Shock and Vibration

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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 7

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(a)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

(b)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(c)

001020304

SCBF

I

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Identified damageInduced damage

(d)

Figure 7 Damage identification for 31-bar truss for case 5 considering (a) 1 mode (b) 2 modes (c) 3 modes and (d) 4 modes

(1)

(3)

(5) (6)

(7) (8)

(9) (10)

(11) (12)

(13) (14)(15) (16)

(17) (18) (19) (20) (21) (22)

(2)

(4)

1

2

3

4

5

6

7

8

9

1011 12

13 1415 1617 18

19 2021 22

29

23 2425 2627

28

3031 32

33

34 35

36

37

38

39 40

41

42

43

44 45

46

47

600 in

540 in

480 in

420 in

360 in

240 in

120 in

0

90 in150 in

30 in

Y

X

60 in 60 in

Figure 8 Forty-seven-bar planar power line tower

8 Shock and Vibration

00102030405

1 5 9 13 17 21 25 29 33 37 41 45

Dam

age r

atio

Element number

(a)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0

02

04

06

(b)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(d)

Figure 9 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 10 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

4 Test Examples

In order to show the capabilities of the proposed method foridentifying the multiple-structural damage two illustrativetest examples are considered The first example is a 31-barplanar truss and the second one is a 47-bar planar trussThe effect of measurement noise on the performance of themethod is considered in the first example

41 Thirty-One-Bar Planar Truss The 31-bar planar trussshown in Figure 2 selected from [23] is modeled using theconventional finite element method without internal nodes

leading to 25 degrees of freedom The material density andelasticity modulus are 2770 kgm3 and 70GPa respectivelyDamage in the structure is simulated as a relative reductionin the elasticity modulus of individual bars Five differentdamage cases given in Table 1 are induced in the structure andthe proposed method is tested for each case Figures 3 4 5 6and 7 show the SCBFI value with respect to element numberfor damage cases 1 to 5 when one to four mode shapes areconsidered respectively

It can be observed that the new index achieves the actualsite of damage truthfully in most cases It is also revealed thatthe general configuration of identification charts does not

Shock and Vibration 9

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

0

02

04

06

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

001020304

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

001020304

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 11 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

00020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 12 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

0005

01015

02025

03

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

001020304

(b)

Figure 13 The SCBFI for 31-bar truss considering 1 noise for (a) case 1 and (b) case 2

10 Shock and Vibration

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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

00102030405

1 5 9 13 17 21 25 29 33 37 41 45

Dam

age r

atio

Element number

(a)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0

02

04

06

(b)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

0020406

SCBFIInduced damageMSEBI

(d)

Figure 9 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 1 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 10 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 2 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

4 Test Examples

In order to show the capabilities of the proposed method foridentifying the multiple-structural damage two illustrativetest examples are considered The first example is a 31-barplanar truss and the second one is a 47-bar planar trussThe effect of measurement noise on the performance of themethod is considered in the first example

41 Thirty-One-Bar Planar Truss The 31-bar planar trussshown in Figure 2 selected from [23] is modeled using theconventional finite element method without internal nodes

leading to 25 degrees of freedom The material density andelasticity modulus are 2770 kgm3 and 70GPa respectivelyDamage in the structure is simulated as a relative reductionin the elasticity modulus of individual bars Five differentdamage cases given in Table 1 are induced in the structure andthe proposed method is tested for each case Figures 3 4 5 6and 7 show the SCBFI value with respect to element numberfor damage cases 1 to 5 when one to four mode shapes areconsidered respectively

It can be observed that the new index achieves the actualsite of damage truthfully in most cases It is also revealed thatthe general configuration of identification charts does not

Shock and Vibration 9

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

0

02

04

06

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

001020304

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

001020304

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 11 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

00020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 12 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

0005

01015

02025

03

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

001020304

(b)

Figure 13 The SCBFI for 31-bar truss considering 1 noise for (a) case 1 and (b) case 2

10 Shock and Vibration

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

0

02

04

06

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

001020304

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

001020304

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 11 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 3 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

00102030405

Dam

age r

atio

1 5 9 13 17 21 25 29 33 37 41 45Element number

(a)

Dam

age r

atio

0

02

04

06

1 5 9 13 17 21 25 29 33 37 41 45Element number

(b)

Dam

age r

atio

0020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(c)

Dam

age r

atio

00020406

1 5 9 13 17 21 25 29 33 37 41 45Element number

SCBFIInduced damageMSEBI

(d)

Figure 12 Comparison of damage index SCBFI with MSEBI for 47-bar planar truss for case 4 considering (a) 1 mode (b) 2 modes (c) 3modes and (d) 4 modes

0005

01015

02025

03

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age r

atio

Element number

Noisy SCBFIInduced damage

001020304

(b)

Figure 13 The SCBFI for 31-bar truss considering 1 noise for (a) case 1 and (b) case 2

10 Shock and Vibration

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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

0

01

02

03D

amag

e rat

io

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

(a)

Dam

age r

atio

001020304

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 14 The SCBFI for 31-bar truss considering 2 noise for (a) case 1 and (b) case 2

0005

01015

02025

03

Dam

age r

atio

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

(a)

001020304

Dam

age r

atio

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Element number

Noisy SCBFIInduced damage

(b)

Figure 15 The SCBFI for 31-bar truss considering 3 noise for (a) case 1 and (b) case 2

change after considering more than twomodes It means thatthe elements detected as damaged elements will be constantvia increasing the measured mode shapes Thus requiringonly two mode shapes for damage localization is one of themost important advantages of the proposed index

42 Forty-Seven-Bar Planar Power Line Tower The 47-barplanar power line tower shown in Figure 8 is the secondexample [24] used to demonstrate the practical capability ofthe new proposed method In this problem the structurehas forty-seven members and twenty-two nodes and is sym-metric about the 119884-axis All members are made of steel andthe material density and modulus of elasticity are 03 lbin3and 30000 ksi respectively Damage in the structure is alsosimulated as a relative reduction in the elasticity modulus ofindividual bars Four different damage cases given in Table 2are induced in the structure and the performance of the newindex (SCBFI) is compared with that of an existing index(MSEBI) [19] Figures 9 10 11 and 12 show the SCBFI andMSEBI values with respect to element number for damagecases 1 to 4 when considering 1 to 4 modes respectively

It is observed that the new proposed index can findthe actual site of damage truthfully Moreover the generalconfiguration of identification charts dose not change whenmore than 3 structural modes are considered It means thatthe elements identified as the damaged elements will beconstant via increasing themode shapesThus requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexibility matrix Moreover thecomparison of the SCBFI with MSEBI index in Figures 9 to

12 demonstrates the same efficiency of the proposed index fordamage localization with respect to MSEBI

43 Analysis of Noise Effect In order to investigate the noiseeffect on the performance of the proposed method themeasurement noise is considered here by an error appliedto the mode shapes [24] Figures 13 14 and 15 show themean value of SCBFI after 100 independent runs for damagecases 1 and 2 of 31-bar truss using 3 mode shapes whenthey randomly polluted through 1 2 and 3 noiserespectively

It can be seen from the Figures 13 and 14 the indexcan localize the damage accurately when 1 and 2 noise respectively are considered however the identificationresults of the method for 3 noise are not appropriate

5 Conclusion

An efficient damage indicator called here as strain changebased on flexibility index (SCBFI) has been proposed forlocating multiple damage cases of truss systems The SCBFIis based on the change of elemental strain computed fromthe flexibility matrix of a structure between the undamagedstructure and damaged structure Since the flexibility matrixused in the calculation of elemental strains converges rapidlywith lower frequencies and mode shapes it will be usefulin decreasing the computational cost In order to assess theperformance of the proposed method for structural damagedetection two illustrative test examples selected from the

Shock and Vibration 11

Table 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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 1 Five different damage cases induced in 31-bar planar truss

Case 1 Case 2 Case 3 Case 4 Case 5Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 025 16 030 1 030 13 025 6 02025 015 mdash mdash 2 020 30 02 15 025mdash mdash mdash mdash mdash mdash mdash mdash 26 030

Table 2 Four different damage cases induced in 47-bar planar power line tower

Case 1 Case 2 Case 3 Case 4Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio27 030 7 030 3 030 25 030mdash mdash 19 020 30 025 38 025mdash mdash mdash mdash 47 020 43 020

literature have been considered The numerical results con-sidering themeasurement noise demonstrate that themethodcan provide an efficient tool for properly locating themultipledamage in the truss systemswhile needing just three vibrationmodes In addition according to the numerical results theindependence of SCBFI with respect to the mode numbers isthe main advantage of the method for damage identificationwithout needing the higher frequencies and mode shapeswhich are practically difficult and experimentally limited tomeasure

Conflict of Interests

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

References

[1] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[2] P Gudmundson ldquoThe dynamic behaviour of slender structureswith cross-sectional cracksrdquo Journal of the Mechanics andPhysics of Solids vol 31 no 4 pp 329ndash345 1983

[3] T K Obrien ldquoStiffness change as a non-destructive damagemeasurementrdquo in Mechanics of Non-Destructive TestIng W WStinchcomb Ed pp 101ndash121 Plenum Press New York NYUSA 1980

[4] P Cawley and R D Adams ldquoThe location of defects instructures from measurements of natural frequenciesrdquo TheJournal of Strain Analysis for Engineering Design vol 14 no 2pp 49ndash57 1979

[5] 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

[6] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[7] M A Mannan and M H Richardson ldquoDetection and locationof structural cracks using FRFmeasurementsrdquo in Proceedings ofthe 8th InternationalModal Analysis Conference (IMAC rsquo90) pp652ndash657 1990

[8] A K Pandey and M Biswas ldquoDamage detection in structuresusing changes in flexibilityrdquo Journal of Sound and Vibration vol169 no 1 pp 3ndash17 1994

[9] M Raghavendrachar and A Aktan ldquoFlexibility by multirefer-ence impact testing for bridge diagnosticsrdquo Journal of StructuralEngineering vol 118 no 8 pp 2186ndash2203 1992

[10] A Tomaszewska ldquoInfluence of statistical errors on damagedetection based on structural flexibility and mode shape cur-vaturerdquo Computers and Structures vol 88 no 3-4 pp 154ndash1642010

[11] J Li BWu Q C Zeng and CW Lim ldquoA generalized flexibilitymatrix based approach for structural damage detectionrdquo Journalof Sound and Vibration vol 329 no 22 pp 4583ndash4587 2010

[12] Q W Yang and B X Sun ldquoStructural damage identificationbased on best achievable flexibility changerdquo Applied Mathemat-ical Modelling vol 35 no 10 pp 5217ndash5224 2011

[13] Q W Yang ldquoA new damage identification method based onstructural flexibility disassemblyrdquo JVCJournal of Vibration andControl vol 17 no 7 pp 1000ndash1008 2011

[14] J Zhang P J Li and Z S Wu ldquoA new flexibility-based damageindex for structural damage detectionrdquo Smart Materials andStructures vol 22 pp 25ndash37 2013

[15] M Nobahari and S M Seyedpoor ldquoAn efficient methodfor structural damage localization based on the concepts offlexibility matrix and strain energy of a structurerdquo StructuralEngineering and Mechanics vol 46 no 2 pp 231ndash244 2013

[16] 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

[17] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[18] H W Shih D P Thambiratnam and T H T Chan ldquoVibrationbased structural damage detection in flexural members usingmulti-criteria approachrdquo Journal of Sound and Vibration vol323 no 3-5 pp 645ndash661 2009

12 Shock and Vibration

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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

[19] S M Seyedpoor ldquoA two stage method for structural damagedetection using a modal strain energy based index and par-ticle swarm optimizationrdquo International Journal of Non-LinearMechanics vol 47 no 1 pp 1ndash8 2012

[20] M Paz andW Leigh Structural Dynamics Theory and Compu-tation Springer 5th edition 2006

[21] R D Cook D S Malkus and M E Plesha Concepts andApplication of Finite Element Analysisedition Wiley New YorkNY USA 3rd edition 1989

[22] D L Logan A First Course in the Finite Element MethodCengage Learning 5th edition 2012

[23] 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

[24] M R N Shirazi H Mollamahmoudi and S M SeyedpoorldquoStructural damage identification using an adaptive multi-stageoptimization method based on a modified particle swarmalgorithmrdquo Journal of Optimization Theory and Applications2013

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