2014 - montazer - damage detection.pdf

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; [email protected]

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

Academic Editor: Gyuhae Park

Copyright 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. Th 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 coeffici ts 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 engineering. Thebasis of many damage identific tion procedures is observ-ing the changes in structural responses. Damage reducesstructures stiffness and mass, which leads to a change inthe static and dynamic responses of the structure. Therefore,the damage detection techniques are generally classifi d intotwo main categories. They include the dynamic and staticidentific tion 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 [13]. Many researchers have used one or more of thesecharacteristics to detect and locate the structural damage.Cawley and Adams [4] used the changes in the naturalfrequencies together with a fin te 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. Thiis 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. The eis 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 locations. The efore, using the changes in mode shapesbetween damaged and undamaged structures may not bevery effici t. Mannan and Richardson utilized the frequencyresponse function (FRF) measurements for detecting andlocating the structural cracks [7]. Pandey and Biswas used thechange in flexi ility matrix to detect damage in a beam [8].Raghavendrachar and Aktan calculated the flexi ility matrixbased on mode shapes and demonstrated the advantagesof using the flexibility compared to the mode shapes [9].The infl ence of statistical errors on damage detection basedon structural flexi ility and mode shape curvature has beeninvestigated by Tomaszewska [10]. Th generalized flexibilitymatrix change to detect the location and extent of structural

Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 460692, 12 pageshttp://dx.doi.org/10.1155/2014/460692

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 achievableflexi ility change with capability in detecting the locationand extent of structural damage has also been presentedby Yang and Sun [12]. A method for structural damageidentific tion based on flexi ility disassembly has also beenproposed by Yang [13]. Th efficie y of the proposed methodhas 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 efficie y 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 elements strains obtained from two setsrelated to intact and damaged structure.The calculation of thestrain is based on a flexi ility matrix estimated from modalanalysis information. Taking advantage of highly convergedflexi ility matrix using only few vibration modes related tolow frequencies in the first phase and then determining theelemental strain using the flexibility coeffici ts 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 fi st described and then the new proposeddamage index is introduced.

2.1.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 coeffici ts 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. Suffici t number of degrees of freedom(number of measurement points) are needed here to attaingood accuracy.

Let [] and [

] be the first and second sets of measured

mode shapes in a matrix form of size

and

,respectively.

and

are the numbers of mode shapes

considered in the respective sets and is the number ofmeasurement points. Th MAC factor is then defin d for themode shapes and as follows:

MAC()=

=1[]

[]

2

=1([]

)2

=1([]

)2, (1)

where = 1, . . . ,

and = 1, . . . ,

; []

and

[]

are the th components of the modes [

] and [

],

respectively. The MAC()

factor indicates the degree ofcorrelation between the th mode of the first set and theth mode of the second set . The MAC values vary from0 to 1, with 0 for no correlation and 1 for full correlation.Therefore, if the eigenvectors [

] and [

] are identical, the

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

Th 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 from two sets of measurements are identical,the COMAC factor is close to 1 for this coordinate. A largedeviation from unity can be interpreted as damage indicationin the structure. For the coordinate of a structure using mode shapes, the COMAC factor is defin d as follows:

COMAC=

=1[]

[]

2

=1([]

)2

=1([]

)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 effici t.

2.2. Flexibility Method. It has been proved that the presenceof damage in a structure increases its flexi ility. 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 damageidentific tion methods is based on using theflexibility matrix.Th flexi ility matrix is the inverse of the stiff ess matrix relating the applied static forces {} to resulting structuraldisplacements {} as

{} = [] {} . (3)

The flexibility matrix can be also dynamically measured frommodal analysis data. Th relationship between the flexi ilitymatrix and the dynamic characteristics of a structure canbe given by [15, 17]:

= [] []1[]

=1

1

2

, (4)

where [] is the mode shape matrix; [] = diag(1/2) isa diagonal matrix;

is the th circular frequency;

is the

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

From (4), one can see that the modal contribution to theflexi ility matrix decreases as the frequency increases. On theother hand, the flexibility matrix converges rapidly with theincrease of the values of the frequencies.Therefore, from onlya few of lower modes, a good estimate for the flexi ility matrixcan be achieved.

Shock and Vibration 3

Th principle of flexibility method is based on a compar-ison of the flexibility matrices from two sets of mode shapes.If

and

are the flexi ility matrices corresponding tothe healthy and damaged states of the structure, a flexibilitychange matrix can be defin d as the difference of the twomatrices:

= . (5)

For each degree of freedom, let

be the maximum absolutevalue of the elements in the corresponding column of :

= max

, (6)

where

are the elements of . In order to identify damage

in the structure, the quantity

can be used as a damageindicator [17].

2.3. Modal Strain Energy Based Method. Th methods basedon modal strain energy of a structure have been commonlyused in damage detection [1719]. 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 identifi ation. Th modal strain energy of ethelement in th mode of the structure can be expressed as [19]:

mse=1

2

= 1, 2, . . . , = 1, 2, . . . , (7)

where is the stiffness matrix of th element of the structureand

is the vector of corresponding nodal displacements of

element in mode .A normalized form of MSE considering vibrating

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

mnmse =

=1(mse/

=1mse)

, = 1, . . . . (8)

Th damage occurrence increases the MSE and consequentlythe effici t parameter mnmse. So, by determining theparameter mnmse for each element of healthy and damagedstructures, an effici t indicator for identifying the damage inthe element can be defin d [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 damage,is developed. Th new index SCBFI proposed is capable ofidentifying and locating the multiple-structural damage intruss structures. Th 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 methods. Th nodal displacement vector usedfor computing the strain obtains from the flexibility matrix of

the structure. Moreover, the flexibility matrix is determinedfrom modal 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 efficie y and lower computational cost.

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

( 2

)

= 0, = 1, 2, . . . , (9)

where and are the stiffness and mass matrices of thestructure, respectively. Also,

and

are the th circular fre-

quency and mode shape vector of the structure, respectively.At 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

=1

1

()2

,

FMD

=1

1

()2

,

(10)

where

and

are the th circular frequency of healthy anddamaged structure, respectively;

and

are the th mode

shape vector of healthy and damaged structure, respectively;and 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. Th mode shapes ofthe damaged structure in partial degrees of freedom arefirs measured, and then the incomplete mode shapes areexpanded to match all degrees of freedom of the structure by acommon technique such as a dynamic condensation method[21].

Since each column of the flexi ility matrix representsthe displacement pattern of the structure, associated with aunit force applied at the corresponding DOF of that column,therefore 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]:

=1

[ cos sin cos sin ]

{{{

{{{

{

1

1

2

2

}}}

}}}

}

, (11)

where is the strain of truss element, is the length ofelement, is the angle between local and global coordinates,and = {

1, 1, 2, 2} is the nodal displacement vector

of the element.


Start

Modal analysis (natural frequencies and mode shapes of

damaged structure)

Calculate the flexi ility 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 flexi ility 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 SMH

Calculate the absolute mean value of each row

of matrix SCM as

SCBFIe = | |

nci=1 SCM

i

e

nc|, e =

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

1, 2, . . . nte

Figur e 1:Th 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

1.52

m

1.52m

Figure 2: Planar truss having 31elements.


00.10.20.30.4

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

SCB

FI

Element number

(a)

0

0.1

0.2

0.3

SCB

FI

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

Element number

(b)

Identified damageInduced damage

0

0.1

0.2

0.3

SCB

FI

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

Element number

(c)


0

0.1

0.2

0.3

SCB

FI

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

Element number

(d)

Figur e 3: Damage identific tion for 31-bar truss for case 1considering (a) 1mode, (b) 2 modes, (c) 3 modes, and (d) 4 modes.

00.10.20.30.4

SCB

FI

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

Element number

(a)

00.10.20.30.4

SCB

FI

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

Element number

(b)

00.10.20.30.4

SCB

FI

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

Element number


(c)

00.10.20.30.4

SCB

FI

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

Element number


(d)

Figur e 4: Damage identific tion for 31-bar truss for case 2 considering (a) 1mode, (b) 2 modes, (c) 3 modes, and (d) 4 modes.

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

SMH=1

[ cos sin cos sin ] {

}

SMD=1

[ cos sin cos sin ] {

} ,

(12)

where SMH

is the strain of th element related to th columnof FMH; SMD

is the strain of th element related to th

column of FMD; {} and {

} are the nodal displacement

vectors of th element, associated with the th column of flexibility matrix for healthy and damaged structure, respectively;

is the angle between local and global coordinates of thelement, and

is the length of th 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 SMH. (13)

Th th row of SCM matrix represents the strain changes of thelement of structure for unit loads applied at different DOFsof the structure. The SCBFI index can now be defin d as acolumnar vector containing the absolute mean value of eachrow of SCM matrix given by

SCBFI=

=1SCM

, = 1, 2, . . . , (14)

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


00.10.20.30.4

SCB

FI

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

Element number

(a)

00.10.20.30.4

SCB

FI

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

Element number

(b)

00.10.20.30.4

SCB

FI

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

Element number


(c)

00.10.20.30.4

SCB

FI

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

Element number


(d)


00.10.20.30.4

SCB

FI

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

Element number

(a)

00.10.20.30.4

SCB

FI

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

Element number

(b)

00.10.20.30.4

SCB

FI

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

Element number


(c)

00.10.20.30.4

SCB

FI

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

Element number


(d)


Th oretically, damage occurrence leads to increasing theSCM and consequently the index SCBFI. As a result, inthis study, by determining the parameter SCBFI for eachelement of the structure, an effici t indicator for estimatingthe presence and locating the damage in the element can bedefin d. Assuming that the set of the damage indices SCBFI

( = 1, 2, . . . ) represents a sample population of a normallydistributed random variable, a normalized damage index canbe defin d as follows:

SCBFI= [(SCBFI

mean (SCBFI))

std (SCBFI)] ,

= 1, 2, . . . ,

(15)

where mean(SCBFI) and std(SCBFI) represent the meanand standard deviation of the vector of damage indices,respectively.

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

SCBFI=

SCBFI

3 SCBFI, = 1, 2, . . . , (16)

where symbolizes the magnitude of a vector.Th process of constructing the SCBFI index can also be

briefl shown in Figure 1.


00.10.20.30.4

SCB

FI

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

Element number

(a)

00.10.20.30.4

SCB

FI

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

Element number

(b)

00.10.20.30.4

SCB

FI

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

Element number


(c)

00.10.20.30.4

SCB

FI

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

Element number


(d)


(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 in.150 in.

30 in.

Y

X

60 in. 60 in.

Figur e 8: Forty-seven-bar planar power line tower.


00.10.20.30.40.5

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

Dam

age

rati

o

Element number

(a)

Dam

age

rati

o

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

Element number

0

0.2

0.4

0.6

(b)

Dam

age

rati

o

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

Element number

0

0.2

0.4

0.6

SCBFIInduced damageMSEBI

(c)

Dam

age

rati

o

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

Element number

0

0.2

0.4

0.6


(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.

00.10.20.30.40.5

Dam

age

rati

o

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

Element number

(a)

Dam

age

rati

o

0

0.2

0.4

0.6

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

(b)

Dam

age

rati

o

0

0.2

0.4

0.6

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

Element number


(c)

Dam

age

rati

o

0

0.2

0.4

0.6

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

Element number


(d)

Figur e 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 truss.Th effect of measurement noise on the performance of themethod is considered in the first example.

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

leading to 25 degrees of freedom. The material density andelasticity modulus are 2770 kg/m3 and 70 GPa, respectively.Damage in the structure is simulated as a relative reductionin the elasticity modulus of individual bars. Five differentdamage cases given in Table 1are induced in the structure andthe proposed method is tested for each case. Figures 3, 4, 5, 6,and 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 configur tion of identific tion charts does not


00.10.20.30.40.5

Dam

age

rati

o

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

Element number

(a)

0

0.2

0.4

0.6

Dam

age

rati

o

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

Element number

(b)

Dam

age

rati

o

00.10.20.30.4

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

Element number


(c)

00.10.20.30.4

Dam

age

rati

o

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

Element number


(d)

Figur e 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.

00.10.20.30.40.5

Dam

age

rati

o

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

Element number

(a)

Dam

age

rati

o

0

0.2

0.4

0.6

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

Element number

(b)

Dam

age

rati

o

0

0.2

0.4

0.6

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

Element number


(c)

Dam

age

rati

o

0.0

0.2

0.4

0.6

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

Element number


(d)

Figur e 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.

00.05

0.10.15

0.20.25

0.3

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

Dam

age

rati

o

Element number

Noisy SCBFIInduced damage

(a)

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

Dam

age

rati

o

Element number


0

0.1

0.2

0.3

0.4

(b)

Figur e 13:The SCBFI for 31-bar truss considering 1%noise for (a) case 1and (b) case 2.


0

0.1

0.2

0.3D

amag

e ra

tio

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

(a)

Dam

age

rati

o

00.10.20.30.4

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

Element number


(b)

Figur e 14: The SCBFI for 31-bar truss considering 2% noise for (a) case 1and (b) case 2.

00.05

0.10.15

0.20.25

0.3

Dam

age

rati

o

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

(a)

0

0.1

0.2

0.3

0.4

Dam

age

rati

o

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

Element number


(b)

Figur e 15:The SCBFI for 31-bar truss considering 3% noise for (a) case 1and (b) case 2.

change after considering more than two modes. 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.

4.2. Forty-Seven-Bar Planar Power Line Tower. Th 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 -axis. All members are made of steel, andthe material density and modulus of elasticity are 0.3 lb/in.3

and 30,000 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 1to 4 when considering 1to 4 modes, respectively.

It is observed that the new proposed index can findthe actual site of damage truthfully. Moreover, the generalconfigur tion of identific tion charts dose not change whenmore than 3 structural modes are considered. It means thatthe elements identified as the damaged elements will beconstant via increasing the mode shapes.Thus, requiring onlythree vibration modes for damage localization is one of themost important advantages of the proposed index that is dueto high convergence of the flexi ility matrix. Moreover, thecomparison of the SCBFI with MSEBI index in Figures 9 to

12demonstrates the same efficie y of the proposed index fordamage localization with respect to MSEBI.

4.3. 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% noise,respectively.

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 identific tionresults of the method for 3% noise are not appropriate.

5. Conclusion

An effici t 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


Ta ble 1: Five different damage cases induced in 31-bar planar truss.

Case 1 Case 2 Case 3 Case 4 Case 5

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

Elementnumber

Damageratio

11 0.25 16 0.30 1 0.30 13 0.25 6 0.20

25 0.15 2 0.20 30 0.2 15 0.25

26 0.30

Ta ble 2: Four different damage cases induced in 47-bar planar power line tower.

Case 1 Case 2 Case 3 Case 4

Element number Damage ratio Element number Damage ratio Element number Damage ratio Element number Damage ratio

27 0.30 7 0.30 3 0.30 25 0.30

19 0.20 30 0.25 38 0.25

47 0.20 43 0.20

literature have been considered. The numerical results con-sidering the measurement noise demonstrate that the methodcan provide an effici t tool for properly locating the multipledamage in the truss systems while 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.

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