Research ArticleDamage Location and Quantification Indices of Shear StructuresBased on Changes in the First Two or Three Natural Frequencies

Hien HoThu and Akira Mita

Mita Laboratory Department of System Design Engineering Keio University 3-14-1 Hiyoshi Kohoku-Ku Yokohama 223-8522 Japan

Correspondence should be addressed to Hien HoThu thuhienjoliegmailcom

Received 15 April 2014 Accepted 23 May 2014 Published 11 June 2014

Academic Editor Constantin Chalioris

Copyright copy 2014 H HoThu and A Mita This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This study proposes damage detection algorithms for multistory shear structures that only need the first two or three naturalfrequencies The methods are able to determine the location and severity of damage on the basis of damage location indices (DLI)and damage quantification indices (DQI) consisting of the changes in the first few squared natural frequencies of the undamagedand damaged statesThe damage is assumed to be represented by a reduction in stiffnessThis stiffness reduction causes a shift in thenatural frequencies of the structureTheuncertainty associatedwith system identificationmethods for obtaining natural frequenciesis also carefully consideredThemethods are accurate and cost-effectivemeans only requiring the changes in the natural frequencies

1 Introduction

Structural health monitoring (SHM) systems are garneringattention as a way of maintaining building structures subjectto natural hazards such as large earthquakes and strongwinds [1] SHM systems play an important role in assessingthe health of a structure because they can determine thelocation and the severity of damage The obstacle we oftenface when installing an SHM system in a building is thetrade-off between the numbers of sensors and the accuracyof damage detection A large number of sensors are costlyand entail a large effort for wiring and installation Suchcomplicated and expensive SHM systems are not feasible formost buildings [2] Therefore a good SHM system shouldhave few sensors yet obtaining enough information about thehealth of a structure

Doebling et al [3] graded SHM systems into four levels ofability as follows

(i) Level 1 determining that damage is present in thestructure

(ii) Level 2 determination of the location of the damage(iii) Level 3 quantification of the severity of the damage(iv) Level 4 prediction of the remaining service life of the

structure

The first three levels are most often related to structuraldynamic testing andmodeling issues Level 4 is not addressedin the structural vibration ormodal analysis literatureHencemost of damage detection methods aim to classify damageinto the first three levels

Damage detection algorithms based on the modal prop-erties of a structure such as modal frequencies modeshapes curvature mode shapes and modal flexibilities havebeen studied in the SHM field for decades However mostalgorithms have difficulties in identifying the precise locationand magnitude of the damage Their accuracy and reliabilityare not considered sufficient if not completely inadequate [2]The key to making a successful damage detection method isthus using a few modal properties of a structure to identifyLevels 1 2 and 3

Zhao and DeWolf [4] presented a sensitivity study com-paring the use of natural frequencies mode shapes andmodal flexibilities for monitoring Based on the fact thatnatural frequencies are sensitive indicators of structuralintegrity the relationship between frequency changes andstructural damage was discussed in a review by Salawu [5]These studies showed that sensitivity analysis of the naturalfrequencies can be a valuable tool in SHM

A damage detectionmethod based on natural frequenciesonly needs two vibration sensors (or even one acceleration

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014 Article ID 459097 8 pageshttpdxdoiorg1011552014459097

2 International Scholarly Research Notices

sensor) to obtain the modal frequencies It is known thatof the various characteristics the natural frequencies are theleast contaminated by measurement noise and can generallybe measured with good accuracy [6] Messina et al [7]suggested a standard error of 015 as a benchmark figurefor natural frequencies measured in the laboratory with theimpulse hammer technique Some researches [8ndash14] haveachieved a standard deviation of less than 1 for the first fewmodes of natural frequencies This low level of error suggeststhat a damage detection method using only information onthe natural frequencies would have acceptable accuracy

Although many previous studies concluded that the fre-quency cannot provide spatial information about structuralchanges multiple frequency shifts may provide spatial infor-mation about structural damage in situations where manynatural frequencies can be measured [15] However thereare only a few natural frequencies that can be measured inmost buildingsTherefore a natural frequency based methodwould need to work with only a few frequencies if it were tobe practical

The purpose of this study is to devise a damage detectionmethod to identify the existence location and amount of

damage to multistory shear structures by using new damageindices consisting of the changes in the first two or threenatural frequencies

2 Sensitivity of Squared Natural FrequencyChanges to Structural Damage

A multistory shear structure (119873-story) can be modeled asa one-dimensional lumped mass shear model as shown inFigure 1 Most of the damage to a structure such as cracksfatigue corrosion and loosening of bolted joints manifestsitself as a stiffness reduction

The characteristic equation for such a structure is writtenas

[K minus 1205962

119903M] 0119903= 0 (1)

where120596119903and 0119903are the 119903th frequency andmode shape vector

respectivelyK is the stiffness matrix (119873 times119873)

K =

[[[[[[[[[[[

[

1198961+ 1198962

minus1198962

0

minus1198962

1198962+ 1198963

minus1198963

0 d d

0

d

dminus119896119894119896119894+ 119896119894+1

minus119896119894+1

d

0 sdot sdot sdot

d d 0

minus119896119873minus1

119896119873minus1

+ 119896119873

minus119896119873

0 minus119896119873

119896119873

]]]]]]]]]]]

]

(2)

andM is the mass matrix (119873 times119873)

M =

[[[[[[[[[[

[

1198981

0

0 1198982

d

0

d 119898

119894

d

0 sdot sdot sdot

d119898119873minus1

0

0 119898119873

]]]]]]]]]]

]

(3)

Solving (1) in the manner of (8) in [15] the generalexpression for the 119894th story stiffness can be obtained as

119896119894=

1205962

119903

(120601119894119903minus 120601(119894minus1)119903

)

119873

sum

119895=119894

119898119895120601119895119903 (4)

where 120601119894119903and 120601

(119894minus1)119903are the 119903th mode shapes at the 119894th and

(119894 minus 1)th stories respectivelyIn addition the squared natural circular frequency of the

119903th mode is expressed in [2] as follows

1205962

119903=

119870119903

119872119903

=120601119879

119903K120601119903

120601119879

119903M120601119903

(5)

where 120596119903is the 119903th natural circular frequency 119870

119903is the 119903th

modal stiffness 119872119903is the 119903th modal mass K and M are the

stiffness and mass matrices and 120601119903is the 119903th mode shape

vectorAlthough the yielded structure is undamaged and may

also have reduced stiffness sometime later in this studywe will assume that the structural damage can be directlyexpressed as a stiffness reduction and that the structural massremains unchanged The difference between the squarednatural frequencies of the 119903th mode of the undamagedstructure and the same structure with damage to the 119894th storyis expressed as

1205962

119903minus 1205962

119903(119894)=120601119879

119903K120601119903

120601119879

119903M120601119903

minus

120601119879

119903(119894)K119894120601119903(119894)

120601119879

119903(119894)M120601119903(119894)

(6)

where the subscript (119894) denotes the 119894th damaged story stateHere 120596

119903and 120596

119903(119894)are the 119903th frequencies of the undam-

aged and 119894th story damaged state K119894is the global stiffness

matrix with the stiffness reduction considered to be a reflec-tion of damage to the 119894th story and is the 119903th mode shapevector of the 119894th damaged story state

In [16] Morita et al gave an equation to obtain the dif-ference in the squared natural frequency of the 119903th mode in

International Scholarly Research Notices 3

(6) by neglecting the difference in mode shape arising fromdamage That difference becomes

1205962

119903minus 1205962

119903(119894)=120601119879

119903ΔK119894120601119903

120601119879

119903M120601119903

(7)

where ΔK119894is the change in the stiffness matrix with the

119894th damaged story When the 119894th story stiffness is reducedonly 119896

(119894minus1)(119894minus1) 119896(119894minus1)119894

119896119894(119894minus1)

and 119896119894119894are altered in the stiffness

matrix ΔK119894is written as

ΔK119894=[[[

[

0 0

119896(119894minus1)(119894minus1)

119896(119894minus1)119894

119896119894(119894minus1)

119896119894119894

0 0

]]]

]

(8)

or

ΔK119894=[[[

[

0 0

Δ119896119894

minusΔ119896119894

minusΔ119896119894

Δ119896119894

0 0

]]]

]

(9)

where Δ119896119894is the change in stiffness of the 119894th story

The squared natural frequency change ratio is determinedby dividing both sides of (7) by 1205962

119903

Δ(119903)

119894=

1205962

119903minus 1205962

119903(119894)

1205962119903

=120601119879

119903ΔK119894120601119903

1205962119903120601119879

119903M120601119903

=120601119879

119903ΔK119894120601119903

120601119879

119903K120601119903

(10)

lArrrArr Δ(119903)

119894=

1

120601119879

119903K120601119903

Δ119896119894(120601119894119903minus 120601(119894minus1)119903

)2

(11)

whereΔ119896119894is the change in stiffness of the 119894th story and 120601

119894119903and

120601(119894minus1)119903

are the 119903th mode shapes at the 119894th and (119894 minus 1)th storyFrom (11) Δ119896

119894is the extent of damage to the 119894th story and

(120601119894119903minus 120601(119894minus1)119903

)2 depends on the location of the 119894th damaged

story As damage normally causes a decrease in the naturalfrequencies Δ(119903)

119894is positive when damage occurs This fre-

quency change ratio Δ(119903)119894may vary depending on the damage

location and quantification Note that for each naturalfrequency there are some locations where the frequency ismost sensitive to the damage while there are other locationswhere the damage has little influence on the frequency

Similarly the frequency change ratio of the 119904thmodeΔ(119904)119894

is obtained as

Δ(119904)

119894=

1

120601119879

119904K120601119904

Δ119896119894(120601119894119904minus 120601(119894minus1)119904

)2

(12)

The sum of these changes is

Δ(119903)

119894+ Δ(119904)

119894= Δ119896119894[

1

120601119879

119903K120601119903

(120601119894119903minus 120601(119894minus1)119903

)2

+1

120601119879

119904K120601119904

(120601119894119904minus 120601(119894minus1)119904

)2

]

(13)

Thus the ratio of the changes in the two natural frequen-cies of the 119903th and 119904th modes can be written as

Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

=

(1120601119879

119903K120601119903) (120601119894119903minus 120601(119894minus1)119903

)2

(1120601119879

119903K120601119903) (120601119894119903minus 120601(119894minus1)119903

)2

+(1120601119879

119904K120601119904) (120601119894119904minus 120601(119894minus1)119904

)2

(14)

997904rArrΔ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

=1

1 + (120601119879

119903K120601119903120601119879

119904K120601119904) ((120601119894119904minus 120601(119894minus1)119904

) (120601119894119903minus 120601(119894minus1)119903

))2

(15)

From (4) we can use the 119903th mode or 119904th mode to obtainthe stiffness of the 119894th story

1205962

119903

(0119894119903minus 0(119894minus1)119903

)

119873

sum

119895=119894

1198981198950119895119903=

1205962

119904

(0119894119904minus 0(119894minus1)119904

)

119873

sum

119895=119894

1198981198950119895119904 (16)

997904rArr120601119894119904minus 120601(119894minus1)119904

120601119894119903minus 120601(119894minus1)119903

=1205962

119904

1205962119903

sum119873

119895=1198941198981198950119895119904

sum119873

119895=1198941198981198950119895119903

(17)

Substituting (17) into (15) we get

Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

=1

1+(120601119879

119903K120601119903120601119879

119904K120601119904) (12059641199041205964119903) (sum119873

119895=1198941198981198950119895119904sum119873

119895=1198941198981198950119895119903)2

(18)

This ratio depends on the location of damage 119894 and thenumber of degrees of freedom in the following coefficient(sum119873

119895=1198941198981198950119895119904sum119873

119895=1198941198981198950119895119903) = (119898

1198940119894119904

+ 119898119894+1

0(119894+1)119904

+ sdot sdot sdot +

1198981198730119873119904)(1198981198940119894119903+119898119894+1

0(119894+1)119903

+sdot sdot sdot+ 1198981198730119873119903) for 119894 = 1 2 119873

3 Proposed Approach to Damage Detection

Let us formulate a simple algorithm for detecting structuraldamage its location and its amount Note that the existenceof damage is determined by a change in the first two or threenatural frequencies In previous some researches [8ndash14] haveachieved a standard deviation of less than 1 for the first fewmodes of natural frequencies In particular we assume thatthe squared natural frequencies have a standard deviationof 1 and damage has occurred when the squared naturalfrequency changes by 2 or more

31 Damage Location Indices The previous section derivedthe ratio of the changes in two squared natural frequenciesof the 119903th and 119904th modes Like (18) this value depends on

4 International Scholarly Research Notices

1st story

mN

kN

mi

ki

m2

k2

m1

k1

Nth story

Nminus 1th story

ith story

Figure 1 Simplified structural model with119873 degrees of freedom

the location of the damageThe damage location index (DLI)is defined as

DLI119903119904 =Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

(19)

The number of obtained frequencies is usually smallerthan the number of degrees of freedom 119873 When two of thefirst two or three frequencies are used the DLI values arewritten as DLI12 DLI23 and so on

DLI12 =Δ(1)

119894

Δ(1)

119894+ Δ(2)

119894

DLI23 =Δ(2)

119894

Δ(2)

119894+ Δ(3)

119894

(20)

The DLI12 value depends on the number of damagedstories 119894 and it can be used as a damage location indicator inthe 119873-story shear structure When taller building has manystories DLI23 andor DLI34 and so forth can be used inaddition to DLI12 to detect the damaged story

32 Uncertainty of Damage Location Indices The modalparameters are often sensitive to various environmentalconditions such as temperature humidity and excitationamplitude The effect of environmental conditions or exci-tation amplitude is treated as ldquonoiserdquo in a simulation so weshould obtain a confidence interval on themodal parameters

Denoting by 1205901205962 the standard deviation of the squared

natural frequency we will discuss the reliability of ourmethod on the basis of the theory presented in [17] We willassume that the squared natural frequencies have a standarddeviation of 1

The standard deviation of the difference between thesquared natural frequencies is

120590(1205962

119903minus1205962

119903(119894))= radic(120590

1205962

119903

)2

+ (1205901205962

119903(119894)

)

2

(21)

Moreover the standard deviation of the changes in the 119903thfrequency Δ(119903) is given by

120590Δ(119903) = Δ

(119903)radic(

120590(1205962

119903minus1205962

119903(119894))

1205962119903minus 1205962

119903(119894)

)

2

+ (

1205901205962

119903

1205962119903

)

2

(22)

The standard deviation of the changes in the 119904th fre-quency 120590

Δ(119904) is calculated similarly to 120590

Δ(119903) in (22)

The standard deviation of the sum of the changes in thefirst two frequencies is

120590(Δ(119903)+Δ(119904))= radic(120590

Δ(119903))2

+ (120590Δ(119904))2

(23)

The standard deviation of the DLI can be calculated as

120590DLI = DLI radic(120590Δ(119903)

Δ(119903))

2

+ (120590(Δ(119903)+Δ(119904))

Δ(119903) + Δ(119904))

2

(24)

Statistically we expect the DLI to be reliable within⟨DLI plusmn 2120590DLI⟩ Hence the confidence interval is 95 and theDLI can identify the damage location in a shear structurewithsatisfactory reliability

33 Damage Quantification Indices Now let us formulate asimple damage quantification technique The change in thenatural frequency can be used to interpolate the amount ofdamage at the detected location [18] Equation (11) also showsthat the change in the natural frequency is proportional tothe stiffness reduction In addition Zhao andDeWolf [4] saidthat the change in natural frequency has a different sensitivitylevel for each location That is some stories show less ofchange in the natural frequency when the damage occurs As[18] we can use the first frequency change Δ(1) to interpolatethe extent of damage However these values are too smallfor some free-end stories that are less sensitive to damageSimilarly the second frequency change Δ

(2) also has lesssensitive locations In these cases it is difficult to interpolatethe extent of damage from the change in only one frequency

The DLI defined in Section 31 was used to detect thelocation of a damaged story in the shear structure NormallyDLI12 is composed of the changes in the first two or threenatural frequencies Equation (7) in [16] can be used toapproximate the extent of damage from the change in anyone of the obtained frequencies A more reliable value can befound by averaging the estimates of all of the modes used inthe analysis

Because the sensitivity to a stiffness reduction depends onthe modes it would be a good idea to introduce the followingaveragingmethodThe damage quantification index (DQI) isdefined by the changes in the first two or three frequenciesthat were used to determine the damaged story

DQI =sum2(3)

119895=1119908119895Δ119896(119895)

119894

sum2(3)

119895=1119908119895

=

1199081Δ119896(1)

119894+ 1199082Δ119896(2)

119894(+1199083Δ119896(3)

119894)

1199081+ 1199082(+1199083)

(25)

where Δ119896(119895)

119894is the stiffness reduction of the damaged story

119894th interpolated by the change in square frequency of the 119895th

International Scholarly Research Notices 5

mode (Δ(119895)

119894) and 119908

119895is a weight with the variance given by

119908119895= 1(120590

Δ(119895))2

The reliability band of Δ119896(119895)119894

is obtained from 120590Δ(119895)

119894

in (22)and the standard deviation of DQI (120590DQI ) is also calculated

4 Performance of Proposed Method

A four-story shear structure was measured to show thefeasibility of the proposed method It was modeled as a one-dimensional lumped mass shear model as shown in Figure 1The damping ratio for each mode was chosen to be 3The sampling frequency 200Hz is commonly used in majormonitoring systems for real buildings However as we canonly need lower natural frequencies the sampling frequencycan be as low as 50Hz provided that an appropriate antialias-ing filter is used In this study the data sampling frequencywas chosen 200Hz The excitation loading is white-noiseAlthough we used white-noise in this paper the excitationcan be any such as winds microtremors and earthquakes

The stories of the structure had the same mass 119898119894=

1000 tons but their stiffness was different 1198961= 119896 = 13 times

103MNm 119896

2= 09119896 119896

3= 08119896 and 119896

4= 07119896 The first two

natural frequencies of the undamaged structure were 189Hzand 519Hz Four cases of damage (damage to the 1st 2nd3rd and 4th stories) were studiedThe damage was simulatedby reducing the stiffness of each story by 5 10 15 2025 and 30

Figure 2 shows the sensitivity of the changes in the firsttwo natural frequencies to the location of the damaged storyFrom Figure 2 it can be seen that the influence of the damagelocation identified by the first natural frequency decreaseson higher stories the results mean that the first naturalfrequency is the most sensitive to damage on the 1st story Bycontrast the second natural frequency is the least sensitive todamage on the 2nd story

Figure 3 uses different colors to plot DLI12 values andtheir confidence interval of 5 to 30 stiffness reduction foreach storyTheDLI12 values of these cases are indicated by thebold lines in the middle of each color their reliability bandconsidering 2120590DLI is indicated by the two thin lines above andbelow each thick line (the standard deviation of the squarednatural frequencies is assumed to be 1)

From Figure 3 we can see the following

(i) The DLI12 values are stable at each damaged story forany level of damage the location of the damaged storycan be detected by examining the correlation betweenthe unknown data and the data of the story on whichdamage was detected

(ii) The variances of 2120590DLI depend on the stiffness reduc-tion when the reduction in stiffness is smaller thesevalues become larger So the extent of damage asindicated by the stiffness reduction must be bigenough to detect the location of damage without anymistakes

In this simulation two modes were enough to detect thedamaged story After determining the location of the damage

Damaged story1 2 3 4

0005

01015

02

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 2nd mode Δ(2)

(a)

1 2 3 40

00501

01502

Damaged story

51015

202530

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 1st mode Δ(1)

(b)

Figure 2 Sensitivity to changes in square natural frequencies (Δ(1)

and Δ(2)) of 4-story structure

0010203040506070809

111

Stiffness reduction ()

2nd story

5 10 15 20 25 30

DLI

12

valu

es an

d th

eir i

nter

vals

damage

1st storydamage3rd storydamage

4th storydamage

Figure 3 DLI12 values and their intervals within the range ⟨DLI plusmn2120590DLI⟩ of 4-story structure

by using the DLI values presented in Figure 3 the quan-tification of damage on that story can be calculated fromthe DQI definition with its standard deviation Figure 4compares the DQI with its uncertainty against real damageto a 4-story structure that reduced the stiffness of each storyby 5 15 and 25

5 Application to Tall Buildings

Next we verified ourmethod on an eight-story structureThedamping ratio was chosen to be 3 and the data samplingfrequency was 200Hz the excitation loading is white-noiseThemass of each story was 1000 tonsThe stiffness of the firststory was assumed to be 119896

1= 119896 = 13 times 10

3MNm and thestiffness of the other stories was 119896

2= 095119896 119896

3= 09119896 119896

4=

085119896 1198965= 08119896 119896

6= 075119896 119896

7= 07119896 and 119896

8= 065119896

6 International Scholarly Research Notices

05

1015202530

Stiff

ness

redu

ctio

n (

)

DQIReal damage

Damage to1st story

Damage to2nd story

Damage to3rd story

Damage to4th story

Figure 4 Comparison of DQI with uncertainty and actual damageof the 4-story structure

Table 1 Damage scenarios of eight-story structure

Damage casenumber

Location of damage(damaged story)

Quantification of damages(percentage of the stiffness

reduction)1 1st 232 4th 143 7th 6

The first three natural frequencies of the undamagedstructure were calculated to be 099 282 and 458Hz Table 1summarizes these three cases in this simulation The firstthree natural frequencies were obtained from accelerationdata of the sensor on top of the simulated structure byapplying the subspace identification method described in[19] The resulting DLI values are displayed in Table 2

Eight cases of damage (from the 1st to 8th story) weresimulated For each case the damage was simulated byreducing the stiffness of each story from5 to 25 and these 21levels of damage were reflected in the changes in the naturalfrequencies The DLI12 values and their reliability band werecalculated by assuming that the squared frequency had astandard deviation of 1 (Figure 5) There is some overlap ofthe 95 confidence intervals The DLI12 values could detectthe location of the damage in three sections of the structureThe DLI12 values of the three damage cases are plotted as thedash dotted lines in Figure 5

The next step is using the 2nd and 3rd frequency changesto get DLI23 and their reliability band Figures 6 7 and 8 plotthe DLI23 values

After determining the location of the damage by usingthe DLI values the amount of damage on that story wascalculated by interpolating the DQI valuesTheDLI and DQIvalues together with their uncertainty in the three damagecases are displayed as the red crosses in Figures 6 to 8

Damage indices are dependent on the mass and stiffnessdistribution Thus we need a prior knowledge of themto apply our method If design drawing is not availablesome system identification tools may be needed This is thelimitation of this method to be applied to a real building

0010203040506070809

111

Stiffness reduction ()5 10 15 20 25

DLI

12

valu

es an

d th

eir i

nter

vals

minus01

Case 2 (095)

3rd and 4th story

Case 1 (051)

1st 2nd and 5th

Case 3 (015)

6th 7th and 8th

damage

story damage

story damage

Figure 5 DLI12 values and their intervals for 8-story structure

040506070809

1

DQI-stiffness reduction ()

Case 1

5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

2nd story damage

1st story damage

5th story damage

Figure 6 DLI23 and DQI values and their intervals for the bluesections

6 Conclusion

The presented method determines the location and amountof damage to shear structures by using the changes in the firstfew natural frequencies only Natural frequencies decrease asa result of damage and the damage indices (DLI andDQI) usethese changes to reveal the location and amount of damageTheDLI uses twonatural frequency changes between undam-aged and damaged states and is stable for each damaged storyAfter identifying the location of the damage the DQI of thedamaged story is used to quantify the damage

As we need only a few natural frequencies two vibrationsensors are enough to obtain the modal frequencies one onthe ground detecting an input and the other on the roofdetecting an output If the input lasts long and the spectrumis flat we may identify those parameters using the outputdata without input informationThus in such a case only onesensor is needed

The uncertainty associated with system identificationmethods for obtaining natural frequencies was also carefullyconsidered and the confidence intervals of the DLI valueswere acceptable with high accuracy The method was alsoshown to be able to detect the location of damage to tallbuildings

Conflict of Interests

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

International Scholarly Research Notices 7

Table 2 Natural frequencies of 3 damage cases of eight-story structure

Damage casenumber

The 1st natural frequency 1205961

[Hz]The 2nd natural frequency 120596

2

[Hz]The 3rd natural frequency 120596

3

[Hz] DLI12 DLI23

1 097 275 447 051 0522 098 282 450 095 0043 099 276 449 015 055

00102030405

Case 2

DQI-stiffness reduction () 5 10 15 20 25minus01

3rd storydamage

4th storydamage

DLI

23

valu

es an

dth

eir i

nter

vals

Figure 7 DLI23 and DQI values and their intervals for the greensection

0203040506070809

111

Case 3

DQI-stiffness reduction () 5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

6th story damage

7th story damage

8th story damage

Figure 8 DLI23 and DQI values and their intervals for the purplesection

Acknowledgments

This work was supported in part by a Grant-in-Aid no22310103 (PI Akira Mita) and Grant-in-Aid to the GlobalCenter of Excellence Program for the ldquoCenter for Educationand Research of Symbiotic Safe and Secure System Designrdquofrom the Ministry of Education Culture Sport Science andTechnology of Japan

References

[1] A Mita ldquoEmerging needs in Japan for health monitoring tech-nologies in civil and building structuresrdquo in Proceedings of the2nd International Workshop on Structural Health Monitoringpp 56ndash67 Stanford University Stanford Calif USA September1999

[2] A Mita Structural Dynamic for Health Monitoring SankeishaNagoya Japan 2003

[3] S W Doebling C R Farrar M B Prime and D W ShevitzDamage Identification and Health Monitoring of Structural andMechanical Systems from Changes in Their Vibration Character-istics A Literature Review Los AlamosNational Laboratory LosAlamos NM USA 1996

[4] J Zhao and J T DeWolf ldquoSensitivity study for vibrationalparameters used in damage detectionrdquo Journal of StructuralEngineering vol 125 no 4 pp 410ndash416 1999

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

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

[7] A Messina I A John and E J Williams ldquoDamage detectionand localization using natural frequency changesrdquo in Proceed-ings of the Conference on Identification in Engineering Systemspp 67ndash76 Swansea UK 1996

[8] P E Carden and A Mita ldquoChallenges in developing confidenceintervals on modal parameters estimated for large civil infras-tructure with stochastic subspace identificationrdquo StructuralControl and Health Monitoring vol 18 no 1 pp 53ndash78 2011

[9] W Gao ldquoInterval natural frequency and mode shape analysisfor truss structures with interval parametersrdquo Finite Elements inAnalysis and Design vol 42 no 6 pp 471ndash477 2006

[10] W Gao ldquoNatural frequency and mode shape analysis ofstructures with uncertaintyrdquo Mechanical Systems and SignalProcessing vol 21 no 1 pp 24ndash39 2007

[11] P Moser and B Moaveni ldquoEnvironmental effects on theidentified natural frequencies of the Dowling Hall FootbridgerdquoMechanical Systems and Signal Processing vol 25 no 7 pp2336ndash2357 2011

[12] K He and W D Zhu ldquoStructural damage detection usingchanges in natural frequencies theory and applicationsrdquo Jour-nal of Physics Conference Series vol 305 no 1 Article ID012054 2011

[13] M D Vila F L Olivares and G R Alcorta ldquoUncertainties inthe estimation of natural frequencies of vibration in buildings ofMexico Cityrdquo Informacion Tecnologica vol 11 no 3 pp 177ndash1842000

[14] J F Clinton S C Bradford T H Heaton and J Favela ldquoTheobserved wander of the natural frequencies in a structurerdquoBulletin of the Seismological Society of America vol 96 no 1pp 237ndash257 2006

[15] J F Wang C C Lin and S M Yen ldquoA story damage indexof seismically-excited buildings based on modal frequency andmode shaperdquo Engineering Structures vol 29 no 9 pp 2143ndash2157 2007

[16] K Morita M Teshigawara and T Hamamoto ldquoDetection andestimation of damage to steel frames through shaking tabletestsrdquo Structural Control and Health Monitoring vol 12 no 3-4pp 357ndash380 2005

[17] V Lindberg Uncertainties and Error Propagation 2008 httpwwwriteducosuphysicsuncertaintiesUncertaintiespart2html

8 International Scholarly Research Notices

[18] H HoThu and A Mita ldquoApplicability of mode-based damageassessment methods to severely damaged steel buildingrdquo inSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 8345 of Proceedings of SPIE SanDiego Calif USA March 2012

[19] M Verhaegen and P Dewilde ldquoSubspace model identificationpart 1 the output-error state-space model identification class ofalgorithmsrdquo International Journal of Control vol 56 no 5 pp1187ndash1210 1992

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International Journal of

2 International Scholarly Research Notices

sensor) to obtain the modal frequencies It is known thatof the various characteristics the natural frequencies are theleast contaminated by measurement noise and can generallybe measured with good accuracy [6] Messina et al [7]suggested a standard error of 015 as a benchmark figurefor natural frequencies measured in the laboratory with theimpulse hammer technique Some researches [8ndash14] haveachieved a standard deviation of less than 1 for the first fewmodes of natural frequencies This low level of error suggeststhat a damage detection method using only information onthe natural frequencies would have acceptable accuracy

Although many previous studies concluded that the fre-quency cannot provide spatial information about structuralchanges multiple frequency shifts may provide spatial infor-mation about structural damage in situations where manynatural frequencies can be measured [15] However thereare only a few natural frequencies that can be measured inmost buildingsTherefore a natural frequency based methodwould need to work with only a few frequencies if it were tobe practical

The purpose of this study is to devise a damage detectionmethod to identify the existence location and amount of

damage to multistory shear structures by using new damageindices consisting of the changes in the first two or threenatural frequencies

2 Sensitivity of Squared Natural FrequencyChanges to Structural Damage

A multistory shear structure (119873-story) can be modeled asa one-dimensional lumped mass shear model as shown inFigure 1 Most of the damage to a structure such as cracksfatigue corrosion and loosening of bolted joints manifestsitself as a stiffness reduction

The characteristic equation for such a structure is writtenas

[K minus 1205962

119903M] 0119903= 0 (1)

where120596119903and 0119903are the 119903th frequency andmode shape vector

respectivelyK is the stiffness matrix (119873 times119873)

K =

[[[[[[[[[[[

[

1198961+ 1198962

minus1198962

0

minus1198962

1198962+ 1198963

minus1198963

0 d d

0

d

dminus119896119894119896119894+ 119896119894+1

minus119896119894+1

d

0 sdot sdot sdot

d d 0

minus119896119873minus1

119896119873minus1

+ 119896119873

minus119896119873

0 minus119896119873

119896119873

]]]]]]]]]]]

]

(2)

andM is the mass matrix (119873 times119873)

M =

[[[[[[[[[[

[

1198981

0

0 1198982

d

0

d 119898

119894

d

0 sdot sdot sdot

d119898119873minus1

0

0 119898119873

]]]]]]]]]]

]

(3)

Solving (1) in the manner of (8) in [15] the generalexpression for the 119894th story stiffness can be obtained as

119896119894=

1205962

119903

(120601119894119903minus 120601(119894minus1)119903

)

119873

sum

119895=119894

119898119895120601119895119903 (4)

where 120601119894119903and 120601

(119894minus1)119903are the 119903th mode shapes at the 119894th and

(119894 minus 1)th stories respectivelyIn addition the squared natural circular frequency of the

119903th mode is expressed in [2] as follows

1205962

119903=

119870119903

119872119903

=120601119879

119903K120601119903

120601119879

119903M120601119903

(5)

where 120596119903is the 119903th natural circular frequency 119870

119903is the 119903th

modal stiffness 119872119903is the 119903th modal mass K and M are the

stiffness and mass matrices and 120601119903is the 119903th mode shape

vectorAlthough the yielded structure is undamaged and may

also have reduced stiffness sometime later in this studywe will assume that the structural damage can be directlyexpressed as a stiffness reduction and that the structural massremains unchanged The difference between the squarednatural frequencies of the 119903th mode of the undamagedstructure and the same structure with damage to the 119894th storyis expressed as

1205962

119903minus 1205962

119903(119894)=120601119879

119903K120601119903

120601119879

119903M120601119903

minus

120601119879

119903(119894)K119894120601119903(119894)

120601119879

119903(119894)M120601119903(119894)

(6)

where the subscript (119894) denotes the 119894th damaged story stateHere 120596

119903and 120596

119903(119894)are the 119903th frequencies of the undam-

aged and 119894th story damaged state K119894is the global stiffness

matrix with the stiffness reduction considered to be a reflec-tion of damage to the 119894th story and is the 119903th mode shapevector of the 119894th damaged story state

In [16] Morita et al gave an equation to obtain the dif-ference in the squared natural frequency of the 119903th mode in

International Scholarly Research Notices 3

(6) by neglecting the difference in mode shape arising fromdamage That difference becomes

1205962

119903minus 1205962

119903(119894)=120601119879

119903ΔK119894120601119903

120601119879

119903M120601119903

(7)

where ΔK119894is the change in the stiffness matrix with the

119894th damaged story When the 119894th story stiffness is reducedonly 119896

(119894minus1)(119894minus1) 119896(119894minus1)119894

119896119894(119894minus1)

and 119896119894119894are altered in the stiffness

matrix ΔK119894is written as

ΔK119894=[[[

[

0 0

119896(119894minus1)(119894minus1)

119896(119894minus1)119894

119896119894(119894minus1)

119896119894119894

0 0

]]]

]

(8)

or

ΔK119894=[[[

[

0 0

Δ119896119894

minusΔ119896119894

minusΔ119896119894

Δ119896119894

0 0

]]]

]

(9)

where Δ119896119894is the change in stiffness of the 119894th story

The squared natural frequency change ratio is determinedby dividing both sides of (7) by 1205962

119903

Δ(119903)

119894=

1205962

119903minus 1205962

119903(119894)

1205962119903

=120601119879

119903ΔK119894120601119903

1205962119903120601119879

119903M120601119903

=120601119879

119903ΔK119894120601119903

120601119879

119903K120601119903

(10)

lArrrArr Δ(119903)

119894=

1

120601119879

119903K120601119903

Δ119896119894(120601119894119903minus 120601(119894minus1)119903

)2

(11)

whereΔ119896119894is the change in stiffness of the 119894th story and 120601

119894119903and

120601(119894minus1)119903

are the 119903th mode shapes at the 119894th and (119894 minus 1)th storyFrom (11) Δ119896

119894is the extent of damage to the 119894th story and

(120601119894119903minus 120601(119894minus1)119903

)2 depends on the location of the 119894th damaged

story As damage normally causes a decrease in the naturalfrequencies Δ(119903)

119894is positive when damage occurs This fre-

quency change ratio Δ(119903)119894may vary depending on the damage

location and quantification Note that for each naturalfrequency there are some locations where the frequency ismost sensitive to the damage while there are other locationswhere the damage has little influence on the frequency

Similarly the frequency change ratio of the 119904thmodeΔ(119904)119894

is obtained as

Δ(119904)

119894=

1

120601119879

119904K120601119904

Δ119896119894(120601119894119904minus 120601(119894minus1)119904

)2

(12)

The sum of these changes is

Δ(119903)

119894+ Δ(119904)

119894= Δ119896119894[

1

120601119879

119903K120601119903

(120601119894119903minus 120601(119894minus1)119903

)2

+1

120601119879

119904K120601119904

(120601119894119904minus 120601(119894minus1)119904

)2

]

(13)

Thus the ratio of the changes in the two natural frequen-cies of the 119903th and 119904th modes can be written as

Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

=

(1120601119879

119903K120601119903) (120601119894119903minus 120601(119894minus1)119903

)2

(1120601119879

119903K120601119903) (120601119894119903minus 120601(119894minus1)119903

)2

+(1120601119879

119904K120601119904) (120601119894119904minus 120601(119894minus1)119904

)2

(14)

997904rArrΔ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

=1

1 + (120601119879

119903K120601119903120601119879

119904K120601119904) ((120601119894119904minus 120601(119894minus1)119904

) (120601119894119903minus 120601(119894minus1)119903

))2

(15)

From (4) we can use the 119903th mode or 119904th mode to obtainthe stiffness of the 119894th story

1205962

119903

(0119894119903minus 0(119894minus1)119903

)

119873

sum

119895=119894

1198981198950119895119903=

1205962

119904

(0119894119904minus 0(119894minus1)119904

)

119873

sum

119895=119894

1198981198950119895119904 (16)

997904rArr120601119894119904minus 120601(119894minus1)119904

120601119894119903minus 120601(119894minus1)119903

=1205962

119904

1205962119903

sum119873

119895=1198941198981198950119895119904

sum119873

119895=1198941198981198950119895119903

(17)

Substituting (17) into (15) we get

Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

=1

1+(120601119879

119903K120601119903120601119879

119904K120601119904) (12059641199041205964119903) (sum119873

119895=1198941198981198950119895119904sum119873

119895=1198941198981198950119895119903)2

(18)

This ratio depends on the location of damage 119894 and thenumber of degrees of freedom in the following coefficient(sum119873

119895=1198941198981198950119895119904sum119873

119895=1198941198981198950119895119903) = (119898

1198940119894119904

+ 119898119894+1

0(119894+1)119904

+ sdot sdot sdot +

1198981198730119873119904)(1198981198940119894119903+119898119894+1

0(119894+1)119903

+sdot sdot sdot+ 1198981198730119873119903) for 119894 = 1 2 119873

3 Proposed Approach to Damage Detection

Let us formulate a simple algorithm for detecting structuraldamage its location and its amount Note that the existenceof damage is determined by a change in the first two or threenatural frequencies In previous some researches [8ndash14] haveachieved a standard deviation of less than 1 for the first fewmodes of natural frequencies In particular we assume thatthe squared natural frequencies have a standard deviationof 1 and damage has occurred when the squared naturalfrequency changes by 2 or more

31 Damage Location Indices The previous section derivedthe ratio of the changes in two squared natural frequenciesof the 119903th and 119904th modes Like (18) this value depends on

4 International Scholarly Research Notices

1st story

mN

kN

mi

ki

m2

k2

m1

k1

Nth story

Nminus 1th story

ith story

Figure 1 Simplified structural model with119873 degrees of freedom

the location of the damageThe damage location index (DLI)is defined as

DLI119903119904 =Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

(19)

The number of obtained frequencies is usually smallerthan the number of degrees of freedom 119873 When two of thefirst two or three frequencies are used the DLI values arewritten as DLI12 DLI23 and so on

DLI12 =Δ(1)

119894

Δ(1)

119894+ Δ(2)

119894

DLI23 =Δ(2)

119894

Δ(2)

119894+ Δ(3)

119894

(20)

The DLI12 value depends on the number of damagedstories 119894 and it can be used as a damage location indicator inthe 119873-story shear structure When taller building has manystories DLI23 andor DLI34 and so forth can be used inaddition to DLI12 to detect the damaged story

32 Uncertainty of Damage Location Indices The modalparameters are often sensitive to various environmentalconditions such as temperature humidity and excitationamplitude The effect of environmental conditions or exci-tation amplitude is treated as ldquonoiserdquo in a simulation so weshould obtain a confidence interval on themodal parameters

Denoting by 1205901205962 the standard deviation of the squared

natural frequency we will discuss the reliability of ourmethod on the basis of the theory presented in [17] We willassume that the squared natural frequencies have a standarddeviation of 1

The standard deviation of the difference between thesquared natural frequencies is

120590(1205962

119903minus1205962

119903(119894))= radic(120590

1205962

119903

)2

+ (1205901205962

119903(119894)

)

2

(21)

Moreover the standard deviation of the changes in the 119903thfrequency Δ(119903) is given by

120590Δ(119903) = Δ

(119903)radic(

120590(1205962

119903minus1205962

119903(119894))

1205962119903minus 1205962

119903(119894)

)

2

+ (

1205901205962

119903

1205962119903

)

2

(22)

The standard deviation of the changes in the 119904th fre-quency 120590

Δ(119904) is calculated similarly to 120590

Δ(119903) in (22)

The standard deviation of the sum of the changes in thefirst two frequencies is

120590(Δ(119903)+Δ(119904))= radic(120590

Δ(119903))2

+ (120590Δ(119904))2

(23)

The standard deviation of the DLI can be calculated as

120590DLI = DLI radic(120590Δ(119903)

Δ(119903))

2

+ (120590(Δ(119903)+Δ(119904))

Δ(119903) + Δ(119904))

2

(24)

Statistically we expect the DLI to be reliable within⟨DLI plusmn 2120590DLI⟩ Hence the confidence interval is 95 and theDLI can identify the damage location in a shear structurewithsatisfactory reliability

33 Damage Quantification Indices Now let us formulate asimple damage quantification technique The change in thenatural frequency can be used to interpolate the amount ofdamage at the detected location [18] Equation (11) also showsthat the change in the natural frequency is proportional tothe stiffness reduction In addition Zhao andDeWolf [4] saidthat the change in natural frequency has a different sensitivitylevel for each location That is some stories show less ofchange in the natural frequency when the damage occurs As[18] we can use the first frequency change Δ(1) to interpolatethe extent of damage However these values are too smallfor some free-end stories that are less sensitive to damageSimilarly the second frequency change Δ

(2) also has lesssensitive locations In these cases it is difficult to interpolatethe extent of damage from the change in only one frequency

The DLI defined in Section 31 was used to detect thelocation of a damaged story in the shear structure NormallyDLI12 is composed of the changes in the first two or threenatural frequencies Equation (7) in [16] can be used toapproximate the extent of damage from the change in anyone of the obtained frequencies A more reliable value can befound by averaging the estimates of all of the modes used inthe analysis

Because the sensitivity to a stiffness reduction depends onthe modes it would be a good idea to introduce the followingaveragingmethodThe damage quantification index (DQI) isdefined by the changes in the first two or three frequenciesthat were used to determine the damaged story

DQI =sum2(3)

119895=1119908119895Δ119896(119895)

119894

sum2(3)

119895=1119908119895

=

1199081Δ119896(1)

119894+ 1199082Δ119896(2)

119894(+1199083Δ119896(3)

119894)

1199081+ 1199082(+1199083)

(25)

where Δ119896(119895)

119894is the stiffness reduction of the damaged story

119894th interpolated by the change in square frequency of the 119895th

International Scholarly Research Notices 5

mode (Δ(119895)

119894) and 119908

119895is a weight with the variance given by

119908119895= 1(120590

Δ(119895))2

The reliability band of Δ119896(119895)119894

is obtained from 120590Δ(119895)

119894

in (22)and the standard deviation of DQI (120590DQI ) is also calculated

4 Performance of Proposed Method

A four-story shear structure was measured to show thefeasibility of the proposed method It was modeled as a one-dimensional lumped mass shear model as shown in Figure 1The damping ratio for each mode was chosen to be 3The sampling frequency 200Hz is commonly used in majormonitoring systems for real buildings However as we canonly need lower natural frequencies the sampling frequencycan be as low as 50Hz provided that an appropriate antialias-ing filter is used In this study the data sampling frequencywas chosen 200Hz The excitation loading is white-noiseAlthough we used white-noise in this paper the excitationcan be any such as winds microtremors and earthquakes

The stories of the structure had the same mass 119898119894=

1000 tons but their stiffness was different 1198961= 119896 = 13 times

103MNm 119896

2= 09119896 119896

3= 08119896 and 119896

4= 07119896 The first two

natural frequencies of the undamaged structure were 189Hzand 519Hz Four cases of damage (damage to the 1st 2nd3rd and 4th stories) were studiedThe damage was simulatedby reducing the stiffness of each story by 5 10 15 2025 and 30

Figure 2 shows the sensitivity of the changes in the firsttwo natural frequencies to the location of the damaged storyFrom Figure 2 it can be seen that the influence of the damagelocation identified by the first natural frequency decreaseson higher stories the results mean that the first naturalfrequency is the most sensitive to damage on the 1st story Bycontrast the second natural frequency is the least sensitive todamage on the 2nd story

Figure 3 uses different colors to plot DLI12 values andtheir confidence interval of 5 to 30 stiffness reduction foreach storyTheDLI12 values of these cases are indicated by thebold lines in the middle of each color their reliability bandconsidering 2120590DLI is indicated by the two thin lines above andbelow each thick line (the standard deviation of the squarednatural frequencies is assumed to be 1)

From Figure 3 we can see the following

(i) The DLI12 values are stable at each damaged story forany level of damage the location of the damaged storycan be detected by examining the correlation betweenthe unknown data and the data of the story on whichdamage was detected

(ii) The variances of 2120590DLI depend on the stiffness reduc-tion when the reduction in stiffness is smaller thesevalues become larger So the extent of damage asindicated by the stiffness reduction must be bigenough to detect the location of damage without anymistakes

In this simulation two modes were enough to detect thedamaged story After determining the location of the damage

Damaged story1 2 3 4

0005

01015

02

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 2nd mode Δ(2)

(a)

1 2 3 40

00501

01502

Damaged story

51015

202530

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 1st mode Δ(1)

(b)

Figure 2 Sensitivity to changes in square natural frequencies (Δ(1)

and Δ(2)) of 4-story structure

0010203040506070809

111

Stiffness reduction ()

2nd story

5 10 15 20 25 30

DLI

12

valu

es an

d th

eir i

nter

vals

damage

1st storydamage3rd storydamage

4th storydamage

Figure 3 DLI12 values and their intervals within the range ⟨DLI plusmn2120590DLI⟩ of 4-story structure

by using the DLI values presented in Figure 3 the quan-tification of damage on that story can be calculated fromthe DQI definition with its standard deviation Figure 4compares the DQI with its uncertainty against real damageto a 4-story structure that reduced the stiffness of each storyby 5 15 and 25

5 Application to Tall Buildings

Next we verified ourmethod on an eight-story structureThedamping ratio was chosen to be 3 and the data samplingfrequency was 200Hz the excitation loading is white-noiseThemass of each story was 1000 tonsThe stiffness of the firststory was assumed to be 119896

1= 119896 = 13 times 10

3MNm and thestiffness of the other stories was 119896

2= 095119896 119896

3= 09119896 119896

4=

085119896 1198965= 08119896 119896

6= 075119896 119896

7= 07119896 and 119896

8= 065119896

6 International Scholarly Research Notices

05

1015202530

Stiff

ness

redu

ctio

n (

)

DQIReal damage

Damage to1st story

Damage to2nd story

Damage to3rd story

Damage to4th story

Figure 4 Comparison of DQI with uncertainty and actual damageof the 4-story structure

Table 1 Damage scenarios of eight-story structure

Damage casenumber

Location of damage(damaged story)

Quantification of damages(percentage of the stiffness

reduction)1 1st 232 4th 143 7th 6

The first three natural frequencies of the undamagedstructure were calculated to be 099 282 and 458Hz Table 1summarizes these three cases in this simulation The firstthree natural frequencies were obtained from accelerationdata of the sensor on top of the simulated structure byapplying the subspace identification method described in[19] The resulting DLI values are displayed in Table 2

Eight cases of damage (from the 1st to 8th story) weresimulated For each case the damage was simulated byreducing the stiffness of each story from5 to 25 and these 21levels of damage were reflected in the changes in the naturalfrequencies The DLI12 values and their reliability band werecalculated by assuming that the squared frequency had astandard deviation of 1 (Figure 5) There is some overlap ofthe 95 confidence intervals The DLI12 values could detectthe location of the damage in three sections of the structureThe DLI12 values of the three damage cases are plotted as thedash dotted lines in Figure 5

The next step is using the 2nd and 3rd frequency changesto get DLI23 and their reliability band Figures 6 7 and 8 plotthe DLI23 values

After determining the location of the damage by usingthe DLI values the amount of damage on that story wascalculated by interpolating the DQI valuesTheDLI and DQIvalues together with their uncertainty in the three damagecases are displayed as the red crosses in Figures 6 to 8

Damage indices are dependent on the mass and stiffnessdistribution Thus we need a prior knowledge of themto apply our method If design drawing is not availablesome system identification tools may be needed This is thelimitation of this method to be applied to a real building

0010203040506070809

111

Stiffness reduction ()5 10 15 20 25

DLI

12

valu

es an

d th

eir i

nter

vals

minus01

Case 2 (095)

3rd and 4th story

Case 1 (051)

1st 2nd and 5th

Case 3 (015)

6th 7th and 8th

damage

story damage

story damage

Figure 5 DLI12 values and their intervals for 8-story structure

040506070809

1

DQI-stiffness reduction ()

Case 1

5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

2nd story damage

1st story damage

5th story damage

Figure 6 DLI23 and DQI values and their intervals for the bluesections

6 Conclusion

The presented method determines the location and amountof damage to shear structures by using the changes in the firstfew natural frequencies only Natural frequencies decrease asa result of damage and the damage indices (DLI andDQI) usethese changes to reveal the location and amount of damageTheDLI uses twonatural frequency changes between undam-aged and damaged states and is stable for each damaged storyAfter identifying the location of the damage the DQI of thedamaged story is used to quantify the damage

As we need only a few natural frequencies two vibrationsensors are enough to obtain the modal frequencies one onthe ground detecting an input and the other on the roofdetecting an output If the input lasts long and the spectrumis flat we may identify those parameters using the outputdata without input informationThus in such a case only onesensor is needed

The uncertainty associated with system identificationmethods for obtaining natural frequencies was also carefullyconsidered and the confidence intervals of the DLI valueswere acceptable with high accuracy The method was alsoshown to be able to detect the location of damage to tallbuildings

Conflict of Interests

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

International Scholarly Research Notices 7

Table 2 Natural frequencies of 3 damage cases of eight-story structure

Damage casenumber

The 1st natural frequency 1205961

[Hz]The 2nd natural frequency 120596

2

[Hz]The 3rd natural frequency 120596

3

[Hz] DLI12 DLI23

1 097 275 447 051 0522 098 282 450 095 0043 099 276 449 015 055

00102030405

Case 2

DQI-stiffness reduction () 5 10 15 20 25minus01

3rd storydamage

4th storydamage

DLI

23

valu

es an

dth

eir i

nter

vals

Figure 7 DLI23 and DQI values and their intervals for the greensection

0203040506070809

111

Case 3

DQI-stiffness reduction () 5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

6th story damage

7th story damage

8th story damage

Figure 8 DLI23 and DQI values and their intervals for the purplesection

Acknowledgments

This work was supported in part by a Grant-in-Aid no22310103 (PI Akira Mita) and Grant-in-Aid to the GlobalCenter of Excellence Program for the ldquoCenter for Educationand Research of Symbiotic Safe and Secure System Designrdquofrom the Ministry of Education Culture Sport Science andTechnology of Japan

References

[1] A Mita ldquoEmerging needs in Japan for health monitoring tech-nologies in civil and building structuresrdquo in Proceedings of the2nd International Workshop on Structural Health Monitoringpp 56ndash67 Stanford University Stanford Calif USA September1999

[2] A Mita Structural Dynamic for Health Monitoring SankeishaNagoya Japan 2003

[3] S W Doebling C R Farrar M B Prime and D W ShevitzDamage Identification and Health Monitoring of Structural andMechanical Systems from Changes in Their Vibration Character-istics A Literature Review Los AlamosNational Laboratory LosAlamos NM USA 1996

[4] J Zhao and J T DeWolf ldquoSensitivity study for vibrationalparameters used in damage detectionrdquo Journal of StructuralEngineering vol 125 no 4 pp 410ndash416 1999

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

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

[7] A Messina I A John and E J Williams ldquoDamage detectionand localization using natural frequency changesrdquo in Proceed-ings of the Conference on Identification in Engineering Systemspp 67ndash76 Swansea UK 1996

[8] P E Carden and A Mita ldquoChallenges in developing confidenceintervals on modal parameters estimated for large civil infras-tructure with stochastic subspace identificationrdquo StructuralControl and Health Monitoring vol 18 no 1 pp 53ndash78 2011

[9] W Gao ldquoInterval natural frequency and mode shape analysisfor truss structures with interval parametersrdquo Finite Elements inAnalysis and Design vol 42 no 6 pp 471ndash477 2006

[10] W Gao ldquoNatural frequency and mode shape analysis ofstructures with uncertaintyrdquo Mechanical Systems and SignalProcessing vol 21 no 1 pp 24ndash39 2007

[11] P Moser and B Moaveni ldquoEnvironmental effects on theidentified natural frequencies of the Dowling Hall FootbridgerdquoMechanical Systems and Signal Processing vol 25 no 7 pp2336ndash2357 2011

[12] K He and W D Zhu ldquoStructural damage detection usingchanges in natural frequencies theory and applicationsrdquo Jour-nal of Physics Conference Series vol 305 no 1 Article ID012054 2011

[13] M D Vila F L Olivares and G R Alcorta ldquoUncertainties inthe estimation of natural frequencies of vibration in buildings ofMexico Cityrdquo Informacion Tecnologica vol 11 no 3 pp 177ndash1842000

[14] J F Clinton S C Bradford T H Heaton and J Favela ldquoTheobserved wander of the natural frequencies in a structurerdquoBulletin of the Seismological Society of America vol 96 no 1pp 237ndash257 2006

[15] J F Wang C C Lin and S M Yen ldquoA story damage indexof seismically-excited buildings based on modal frequency andmode shaperdquo Engineering Structures vol 29 no 9 pp 2143ndash2157 2007

[16] K Morita M Teshigawara and T Hamamoto ldquoDetection andestimation of damage to steel frames through shaking tabletestsrdquo Structural Control and Health Monitoring vol 12 no 3-4pp 357ndash380 2005

[17] V Lindberg Uncertainties and Error Propagation 2008 httpwwwriteducosuphysicsuncertaintiesUncertaintiespart2html

8 International Scholarly Research Notices

[18] H HoThu and A Mita ldquoApplicability of mode-based damageassessment methods to severely damaged steel buildingrdquo inSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 8345 of Proceedings of SPIE SanDiego Calif USA March 2012

[19] M Verhaegen and P Dewilde ldquoSubspace model identificationpart 1 the output-error state-space model identification class ofalgorithmsrdquo International Journal of Control vol 56 no 5 pp1187ndash1210 1992

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International Journal of

International Scholarly Research Notices 3

(6) by neglecting the difference in mode shape arising fromdamage That difference becomes

1205962

119903minus 1205962

119903(119894)=120601119879

119903ΔK119894120601119903

120601119879

119903M120601119903

(7)

where ΔK119894is the change in the stiffness matrix with the

119894th damaged story When the 119894th story stiffness is reducedonly 119896

(119894minus1)(119894minus1) 119896(119894minus1)119894

119896119894(119894minus1)

and 119896119894119894are altered in the stiffness

matrix ΔK119894is written as

ΔK119894=[[[

[

0 0

119896(119894minus1)(119894minus1)

119896(119894minus1)119894

119896119894(119894minus1)

119896119894119894

0 0

]]]

]

(8)

or

ΔK119894=[[[

[

0 0

Δ119896119894

minusΔ119896119894

minusΔ119896119894

Δ119896119894

0 0

]]]

]

(9)

where Δ119896119894is the change in stiffness of the 119894th story

The squared natural frequency change ratio is determinedby dividing both sides of (7) by 1205962

119903

Δ(119903)

119894=

1205962

119903minus 1205962

119903(119894)

1205962119903

=120601119879

119903ΔK119894120601119903

1205962119903120601119879

119903M120601119903

=120601119879

119903ΔK119894120601119903

120601119879

119903K120601119903

(10)

lArrrArr Δ(119903)

119894=

1

120601119879

119903K120601119903

Δ119896119894(120601119894119903minus 120601(119894minus1)119903

)2

(11)

whereΔ119896119894is the change in stiffness of the 119894th story and 120601

119894119903and

120601(119894minus1)119903

are the 119903th mode shapes at the 119894th and (119894 minus 1)th storyFrom (11) Δ119896

119894is the extent of damage to the 119894th story and

(120601119894119903minus 120601(119894minus1)119903

)2 depends on the location of the 119894th damaged

story As damage normally causes a decrease in the naturalfrequencies Δ(119903)

119894is positive when damage occurs This fre-

quency change ratio Δ(119903)119894may vary depending on the damage

location and quantification Note that for each naturalfrequency there are some locations where the frequency ismost sensitive to the damage while there are other locationswhere the damage has little influence on the frequency

Similarly the frequency change ratio of the 119904thmodeΔ(119904)119894

is obtained as

Δ(119904)

119894=

1

120601119879

119904K120601119904

Δ119896119894(120601119894119904minus 120601(119894minus1)119904

)2

(12)

The sum of these changes is

Δ(119903)

119894+ Δ(119904)

119894= Δ119896119894[

1

120601119879

119903K120601119903

(120601119894119903minus 120601(119894minus1)119903

)2

+1

120601119879

119904K120601119904

(120601119894119904minus 120601(119894minus1)119904

)2

]

(13)

Thus the ratio of the changes in the two natural frequen-cies of the 119903th and 119904th modes can be written as

Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

=

(1120601119879

119903K120601119903) (120601119894119903minus 120601(119894minus1)119903

)2

(1120601119879

119903K120601119903) (120601119894119903minus 120601(119894minus1)119903

)2

+(1120601119879

119904K120601119904) (120601119894119904minus 120601(119894minus1)119904

)2

(14)

997904rArrΔ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

=1

1 + (120601119879

119903K120601119903120601119879

119904K120601119904) ((120601119894119904minus 120601(119894minus1)119904

) (120601119894119903minus 120601(119894minus1)119903

))2

(15)

From (4) we can use the 119903th mode or 119904th mode to obtainthe stiffness of the 119894th story

1205962

119903

(0119894119903minus 0(119894minus1)119903

)

119873

sum

119895=119894

1198981198950119895119903=

1205962

119904

(0119894119904minus 0(119894minus1)119904

)

119873

sum

119895=119894

1198981198950119895119904 (16)

997904rArr120601119894119904minus 120601(119894minus1)119904

120601119894119903minus 120601(119894minus1)119903

=1205962

119904

1205962119903

sum119873

119895=1198941198981198950119895119904

sum119873

119895=1198941198981198950119895119903

(17)

Substituting (17) into (15) we get

Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

=1

1+(120601119879

119903K120601119903120601119879

119904K120601119904) (12059641199041205964119903) (sum119873

119895=1198941198981198950119895119904sum119873

119895=1198941198981198950119895119903)2

(18)

This ratio depends on the location of damage 119894 and thenumber of degrees of freedom in the following coefficient(sum119873

119895=1198941198981198950119895119904sum119873

119895=1198941198981198950119895119903) = (119898

1198940119894119904

+ 119898119894+1

0(119894+1)119904

+ sdot sdot sdot +

1198981198730119873119904)(1198981198940119894119903+119898119894+1

0(119894+1)119903

+sdot sdot sdot+ 1198981198730119873119903) for 119894 = 1 2 119873

3 Proposed Approach to Damage Detection

Let us formulate a simple algorithm for detecting structuraldamage its location and its amount Note that the existenceof damage is determined by a change in the first two or threenatural frequencies In previous some researches [8ndash14] haveachieved a standard deviation of less than 1 for the first fewmodes of natural frequencies In particular we assume thatthe squared natural frequencies have a standard deviationof 1 and damage has occurred when the squared naturalfrequency changes by 2 or more

31 Damage Location Indices The previous section derivedthe ratio of the changes in two squared natural frequenciesof the 119903th and 119904th modes Like (18) this value depends on

4 International Scholarly Research Notices

1st story

mN

kN

mi

ki

m2

k2

m1

k1

Nth story

Nminus 1th story

ith story

Figure 1 Simplified structural model with119873 degrees of freedom

the location of the damageThe damage location index (DLI)is defined as

DLI119903119904 =Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

(19)

The number of obtained frequencies is usually smallerthan the number of degrees of freedom 119873 When two of thefirst two or three frequencies are used the DLI values arewritten as DLI12 DLI23 and so on

DLI12 =Δ(1)

119894

Δ(1)

119894+ Δ(2)

119894

DLI23 =Δ(2)

119894

Δ(2)

119894+ Δ(3)

119894

(20)

The DLI12 value depends on the number of damagedstories 119894 and it can be used as a damage location indicator inthe 119873-story shear structure When taller building has manystories DLI23 andor DLI34 and so forth can be used inaddition to DLI12 to detect the damaged story

32 Uncertainty of Damage Location Indices The modalparameters are often sensitive to various environmentalconditions such as temperature humidity and excitationamplitude The effect of environmental conditions or exci-tation amplitude is treated as ldquonoiserdquo in a simulation so weshould obtain a confidence interval on themodal parameters

Denoting by 1205901205962 the standard deviation of the squared

natural frequency we will discuss the reliability of ourmethod on the basis of the theory presented in [17] We willassume that the squared natural frequencies have a standarddeviation of 1

The standard deviation of the difference between thesquared natural frequencies is

120590(1205962

119903minus1205962

119903(119894))= radic(120590

1205962

119903

)2

+ (1205901205962

119903(119894)

)

2

(21)

Moreover the standard deviation of the changes in the 119903thfrequency Δ(119903) is given by

120590Δ(119903) = Δ

(119903)radic(

120590(1205962

119903minus1205962

119903(119894))

1205962119903minus 1205962

119903(119894)

)

2

+ (

1205901205962

119903

1205962119903

)

2

(22)

The standard deviation of the changes in the 119904th fre-quency 120590

Δ(119904) is calculated similarly to 120590

Δ(119903) in (22)

The standard deviation of the sum of the changes in thefirst two frequencies is

120590(Δ(119903)+Δ(119904))= radic(120590

Δ(119903))2

+ (120590Δ(119904))2

(23)

The standard deviation of the DLI can be calculated as

120590DLI = DLI radic(120590Δ(119903)

Δ(119903))

2

+ (120590(Δ(119903)+Δ(119904))

Δ(119903) + Δ(119904))

2

(24)

Statistically we expect the DLI to be reliable within⟨DLI plusmn 2120590DLI⟩ Hence the confidence interval is 95 and theDLI can identify the damage location in a shear structurewithsatisfactory reliability

33 Damage Quantification Indices Now let us formulate asimple damage quantification technique The change in thenatural frequency can be used to interpolate the amount ofdamage at the detected location [18] Equation (11) also showsthat the change in the natural frequency is proportional tothe stiffness reduction In addition Zhao andDeWolf [4] saidthat the change in natural frequency has a different sensitivitylevel for each location That is some stories show less ofchange in the natural frequency when the damage occurs As[18] we can use the first frequency change Δ(1) to interpolatethe extent of damage However these values are too smallfor some free-end stories that are less sensitive to damageSimilarly the second frequency change Δ

(2) also has lesssensitive locations In these cases it is difficult to interpolatethe extent of damage from the change in only one frequency

The DLI defined in Section 31 was used to detect thelocation of a damaged story in the shear structure NormallyDLI12 is composed of the changes in the first two or threenatural frequencies Equation (7) in [16] can be used toapproximate the extent of damage from the change in anyone of the obtained frequencies A more reliable value can befound by averaging the estimates of all of the modes used inthe analysis

Because the sensitivity to a stiffness reduction depends onthe modes it would be a good idea to introduce the followingaveragingmethodThe damage quantification index (DQI) isdefined by the changes in the first two or three frequenciesthat were used to determine the damaged story

DQI =sum2(3)

119895=1119908119895Δ119896(119895)

119894

sum2(3)

119895=1119908119895

=

1199081Δ119896(1)

119894+ 1199082Δ119896(2)

119894(+1199083Δ119896(3)

119894)

1199081+ 1199082(+1199083)

(25)

where Δ119896(119895)

119894is the stiffness reduction of the damaged story

119894th interpolated by the change in square frequency of the 119895th

International Scholarly Research Notices 5

mode (Δ(119895)

119894) and 119908

119895is a weight with the variance given by

119908119895= 1(120590

Δ(119895))2

The reliability band of Δ119896(119895)119894

is obtained from 120590Δ(119895)

119894

in (22)and the standard deviation of DQI (120590DQI ) is also calculated

4 Performance of Proposed Method

A four-story shear structure was measured to show thefeasibility of the proposed method It was modeled as a one-dimensional lumped mass shear model as shown in Figure 1The damping ratio for each mode was chosen to be 3The sampling frequency 200Hz is commonly used in majormonitoring systems for real buildings However as we canonly need lower natural frequencies the sampling frequencycan be as low as 50Hz provided that an appropriate antialias-ing filter is used In this study the data sampling frequencywas chosen 200Hz The excitation loading is white-noiseAlthough we used white-noise in this paper the excitationcan be any such as winds microtremors and earthquakes

The stories of the structure had the same mass 119898119894=

1000 tons but their stiffness was different 1198961= 119896 = 13 times

103MNm 119896

2= 09119896 119896

3= 08119896 and 119896

4= 07119896 The first two

natural frequencies of the undamaged structure were 189Hzand 519Hz Four cases of damage (damage to the 1st 2nd3rd and 4th stories) were studiedThe damage was simulatedby reducing the stiffness of each story by 5 10 15 2025 and 30

Figure 2 shows the sensitivity of the changes in the firsttwo natural frequencies to the location of the damaged storyFrom Figure 2 it can be seen that the influence of the damagelocation identified by the first natural frequency decreaseson higher stories the results mean that the first naturalfrequency is the most sensitive to damage on the 1st story Bycontrast the second natural frequency is the least sensitive todamage on the 2nd story

Figure 3 uses different colors to plot DLI12 values andtheir confidence interval of 5 to 30 stiffness reduction foreach storyTheDLI12 values of these cases are indicated by thebold lines in the middle of each color their reliability bandconsidering 2120590DLI is indicated by the two thin lines above andbelow each thick line (the standard deviation of the squarednatural frequencies is assumed to be 1)

From Figure 3 we can see the following

(i) The DLI12 values are stable at each damaged story forany level of damage the location of the damaged storycan be detected by examining the correlation betweenthe unknown data and the data of the story on whichdamage was detected

(ii) The variances of 2120590DLI depend on the stiffness reduc-tion when the reduction in stiffness is smaller thesevalues become larger So the extent of damage asindicated by the stiffness reduction must be bigenough to detect the location of damage without anymistakes

In this simulation two modes were enough to detect thedamaged story After determining the location of the damage

Damaged story1 2 3 4

0005

01015

02

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 2nd mode Δ(2)

(a)

1 2 3 40

00501

01502

Damaged story

51015

202530

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 1st mode Δ(1)

(b)

Figure 2 Sensitivity to changes in square natural frequencies (Δ(1)

and Δ(2)) of 4-story structure

0010203040506070809

111

Stiffness reduction ()

2nd story

5 10 15 20 25 30

DLI

12

valu

es an

d th

eir i

nter

vals

damage

1st storydamage3rd storydamage

4th storydamage

Figure 3 DLI12 values and their intervals within the range ⟨DLI plusmn2120590DLI⟩ of 4-story structure

by using the DLI values presented in Figure 3 the quan-tification of damage on that story can be calculated fromthe DQI definition with its standard deviation Figure 4compares the DQI with its uncertainty against real damageto a 4-story structure that reduced the stiffness of each storyby 5 15 and 25

5 Application to Tall Buildings

Next we verified ourmethod on an eight-story structureThedamping ratio was chosen to be 3 and the data samplingfrequency was 200Hz the excitation loading is white-noiseThemass of each story was 1000 tonsThe stiffness of the firststory was assumed to be 119896

1= 119896 = 13 times 10

3MNm and thestiffness of the other stories was 119896

2= 095119896 119896

3= 09119896 119896

4=

085119896 1198965= 08119896 119896

6= 075119896 119896

7= 07119896 and 119896

8= 065119896

6 International Scholarly Research Notices

05

1015202530

Stiff

ness

redu

ctio

n (

)

DQIReal damage

Damage to1st story

Damage to2nd story

Damage to3rd story

Damage to4th story

Figure 4 Comparison of DQI with uncertainty and actual damageof the 4-story structure

Table 1 Damage scenarios of eight-story structure

Damage casenumber

Location of damage(damaged story)

Quantification of damages(percentage of the stiffness

reduction)1 1st 232 4th 143 7th 6

The first three natural frequencies of the undamagedstructure were calculated to be 099 282 and 458Hz Table 1summarizes these three cases in this simulation The firstthree natural frequencies were obtained from accelerationdata of the sensor on top of the simulated structure byapplying the subspace identification method described in[19] The resulting DLI values are displayed in Table 2

Eight cases of damage (from the 1st to 8th story) weresimulated For each case the damage was simulated byreducing the stiffness of each story from5 to 25 and these 21levels of damage were reflected in the changes in the naturalfrequencies The DLI12 values and their reliability band werecalculated by assuming that the squared frequency had astandard deviation of 1 (Figure 5) There is some overlap ofthe 95 confidence intervals The DLI12 values could detectthe location of the damage in three sections of the structureThe DLI12 values of the three damage cases are plotted as thedash dotted lines in Figure 5

The next step is using the 2nd and 3rd frequency changesto get DLI23 and their reliability band Figures 6 7 and 8 plotthe DLI23 values

After determining the location of the damage by usingthe DLI values the amount of damage on that story wascalculated by interpolating the DQI valuesTheDLI and DQIvalues together with their uncertainty in the three damagecases are displayed as the red crosses in Figures 6 to 8

Damage indices are dependent on the mass and stiffnessdistribution Thus we need a prior knowledge of themto apply our method If design drawing is not availablesome system identification tools may be needed This is thelimitation of this method to be applied to a real building

0010203040506070809

111

Stiffness reduction ()5 10 15 20 25

DLI

12

valu

es an

d th

eir i

nter

vals

minus01

Case 2 (095)

3rd and 4th story

Case 1 (051)

1st 2nd and 5th

Case 3 (015)

6th 7th and 8th

damage

story damage

story damage

Figure 5 DLI12 values and their intervals for 8-story structure

040506070809

1

DQI-stiffness reduction ()

Case 1

5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

2nd story damage

1st story damage

5th story damage

Figure 6 DLI23 and DQI values and their intervals for the bluesections

6 Conclusion

The presented method determines the location and amountof damage to shear structures by using the changes in the firstfew natural frequencies only Natural frequencies decrease asa result of damage and the damage indices (DLI andDQI) usethese changes to reveal the location and amount of damageTheDLI uses twonatural frequency changes between undam-aged and damaged states and is stable for each damaged storyAfter identifying the location of the damage the DQI of thedamaged story is used to quantify the damage

As we need only a few natural frequencies two vibrationsensors are enough to obtain the modal frequencies one onthe ground detecting an input and the other on the roofdetecting an output If the input lasts long and the spectrumis flat we may identify those parameters using the outputdata without input informationThus in such a case only onesensor is needed

The uncertainty associated with system identificationmethods for obtaining natural frequencies was also carefullyconsidered and the confidence intervals of the DLI valueswere acceptable with high accuracy The method was alsoshown to be able to detect the location of damage to tallbuildings

Conflict of Interests

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

International Scholarly Research Notices 7

Table 2 Natural frequencies of 3 damage cases of eight-story structure

Damage casenumber

The 1st natural frequency 1205961

[Hz]The 2nd natural frequency 120596

2

[Hz]The 3rd natural frequency 120596

3

[Hz] DLI12 DLI23

1 097 275 447 051 0522 098 282 450 095 0043 099 276 449 015 055

00102030405

Case 2

DQI-stiffness reduction () 5 10 15 20 25minus01

3rd storydamage

4th storydamage

DLI

23

valu

es an

dth

eir i

nter

vals

Figure 7 DLI23 and DQI values and their intervals for the greensection

0203040506070809

111

Case 3

DQI-stiffness reduction () 5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

6th story damage

7th story damage

8th story damage

Figure 8 DLI23 and DQI values and their intervals for the purplesection

Acknowledgments

This work was supported in part by a Grant-in-Aid no22310103 (PI Akira Mita) and Grant-in-Aid to the GlobalCenter of Excellence Program for the ldquoCenter for Educationand Research of Symbiotic Safe and Secure System Designrdquofrom the Ministry of Education Culture Sport Science andTechnology of Japan

References

[1] A Mita ldquoEmerging needs in Japan for health monitoring tech-nologies in civil and building structuresrdquo in Proceedings of the2nd International Workshop on Structural Health Monitoringpp 56ndash67 Stanford University Stanford Calif USA September1999

[2] A Mita Structural Dynamic for Health Monitoring SankeishaNagoya Japan 2003

[3] S W Doebling C R Farrar M B Prime and D W ShevitzDamage Identification and Health Monitoring of Structural andMechanical Systems from Changes in Their Vibration Character-istics A Literature Review Los AlamosNational Laboratory LosAlamos NM USA 1996

[4] J Zhao and J T DeWolf ldquoSensitivity study for vibrationalparameters used in damage detectionrdquo Journal of StructuralEngineering vol 125 no 4 pp 410ndash416 1999

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

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

[7] A Messina I A John and E J Williams ldquoDamage detectionand localization using natural frequency changesrdquo in Proceed-ings of the Conference on Identification in Engineering Systemspp 67ndash76 Swansea UK 1996

[8] P E Carden and A Mita ldquoChallenges in developing confidenceintervals on modal parameters estimated for large civil infras-tructure with stochastic subspace identificationrdquo StructuralControl and Health Monitoring vol 18 no 1 pp 53ndash78 2011

[9] W Gao ldquoInterval natural frequency and mode shape analysisfor truss structures with interval parametersrdquo Finite Elements inAnalysis and Design vol 42 no 6 pp 471ndash477 2006

[10] W Gao ldquoNatural frequency and mode shape analysis ofstructures with uncertaintyrdquo Mechanical Systems and SignalProcessing vol 21 no 1 pp 24ndash39 2007

[11] P Moser and B Moaveni ldquoEnvironmental effects on theidentified natural frequencies of the Dowling Hall FootbridgerdquoMechanical Systems and Signal Processing vol 25 no 7 pp2336ndash2357 2011

[12] K He and W D Zhu ldquoStructural damage detection usingchanges in natural frequencies theory and applicationsrdquo Jour-nal of Physics Conference Series vol 305 no 1 Article ID012054 2011

[13] M D Vila F L Olivares and G R Alcorta ldquoUncertainties inthe estimation of natural frequencies of vibration in buildings ofMexico Cityrdquo Informacion Tecnologica vol 11 no 3 pp 177ndash1842000

[14] J F Clinton S C Bradford T H Heaton and J Favela ldquoTheobserved wander of the natural frequencies in a structurerdquoBulletin of the Seismological Society of America vol 96 no 1pp 237ndash257 2006

[15] J F Wang C C Lin and S M Yen ldquoA story damage indexof seismically-excited buildings based on modal frequency andmode shaperdquo Engineering Structures vol 29 no 9 pp 2143ndash2157 2007

[16] K Morita M Teshigawara and T Hamamoto ldquoDetection andestimation of damage to steel frames through shaking tabletestsrdquo Structural Control and Health Monitoring vol 12 no 3-4pp 357ndash380 2005

[17] V Lindberg Uncertainties and Error Propagation 2008 httpwwwriteducosuphysicsuncertaintiesUncertaintiespart2html

8 International Scholarly Research Notices

[18] H HoThu and A Mita ldquoApplicability of mode-based damageassessment methods to severely damaged steel buildingrdquo inSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 8345 of Proceedings of SPIE SanDiego Calif USA March 2012

[19] M Verhaegen and P Dewilde ldquoSubspace model identificationpart 1 the output-error state-space model identification class ofalgorithmsrdquo International Journal of Control vol 56 no 5 pp1187ndash1210 1992

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

4 International Scholarly Research Notices

1st story

mN

kN

mi

ki

m2

k2

m1

k1

Nth story

Nminus 1th story

ith story

Figure 1 Simplified structural model with119873 degrees of freedom

the location of the damageThe damage location index (DLI)is defined as

DLI119903119904 =Δ(119903)

119894

Δ(119903)

119894+ Δ(119904)

119894

(19)

The number of obtained frequencies is usually smallerthan the number of degrees of freedom 119873 When two of thefirst two or three frequencies are used the DLI values arewritten as DLI12 DLI23 and so on

DLI12 =Δ(1)

119894

Δ(1)

119894+ Δ(2)

119894

DLI23 =Δ(2)

119894

Δ(2)

119894+ Δ(3)

119894

(20)

The DLI12 value depends on the number of damagedstories 119894 and it can be used as a damage location indicator inthe 119873-story shear structure When taller building has manystories DLI23 andor DLI34 and so forth can be used inaddition to DLI12 to detect the damaged story

32 Uncertainty of Damage Location Indices The modalparameters are often sensitive to various environmentalconditions such as temperature humidity and excitationamplitude The effect of environmental conditions or exci-tation amplitude is treated as ldquonoiserdquo in a simulation so weshould obtain a confidence interval on themodal parameters

Denoting by 1205901205962 the standard deviation of the squared

natural frequency we will discuss the reliability of ourmethod on the basis of the theory presented in [17] We willassume that the squared natural frequencies have a standarddeviation of 1

The standard deviation of the difference between thesquared natural frequencies is

120590(1205962

119903minus1205962

119903(119894))= radic(120590

1205962

119903

)2

+ (1205901205962

119903(119894)

)

2

(21)

Moreover the standard deviation of the changes in the 119903thfrequency Δ(119903) is given by

120590Δ(119903) = Δ

(119903)radic(

120590(1205962

119903minus1205962

119903(119894))

1205962119903minus 1205962

119903(119894)

)

2

+ (

1205901205962

119903

1205962119903

)

2

(22)

The standard deviation of the changes in the 119904th fre-quency 120590

Δ(119904) is calculated similarly to 120590

Δ(119903) in (22)

The standard deviation of the sum of the changes in thefirst two frequencies is

120590(Δ(119903)+Δ(119904))= radic(120590

Δ(119903))2

+ (120590Δ(119904))2

(23)

The standard deviation of the DLI can be calculated as

120590DLI = DLI radic(120590Δ(119903)

Δ(119903))

2

+ (120590(Δ(119903)+Δ(119904))

Δ(119903) + Δ(119904))

2

(24)

Statistically we expect the DLI to be reliable within⟨DLI plusmn 2120590DLI⟩ Hence the confidence interval is 95 and theDLI can identify the damage location in a shear structurewithsatisfactory reliability

33 Damage Quantification Indices Now let us formulate asimple damage quantification technique The change in thenatural frequency can be used to interpolate the amount ofdamage at the detected location [18] Equation (11) also showsthat the change in the natural frequency is proportional tothe stiffness reduction In addition Zhao andDeWolf [4] saidthat the change in natural frequency has a different sensitivitylevel for each location That is some stories show less ofchange in the natural frequency when the damage occurs As[18] we can use the first frequency change Δ(1) to interpolatethe extent of damage However these values are too smallfor some free-end stories that are less sensitive to damageSimilarly the second frequency change Δ

(2) also has lesssensitive locations In these cases it is difficult to interpolatethe extent of damage from the change in only one frequency

The DLI defined in Section 31 was used to detect thelocation of a damaged story in the shear structure NormallyDLI12 is composed of the changes in the first two or threenatural frequencies Equation (7) in [16] can be used toapproximate the extent of damage from the change in anyone of the obtained frequencies A more reliable value can befound by averaging the estimates of all of the modes used inthe analysis

Because the sensitivity to a stiffness reduction depends onthe modes it would be a good idea to introduce the followingaveragingmethodThe damage quantification index (DQI) isdefined by the changes in the first two or three frequenciesthat were used to determine the damaged story

DQI =sum2(3)

119895=1119908119895Δ119896(119895)

119894

sum2(3)

119895=1119908119895

=

1199081Δ119896(1)

119894+ 1199082Δ119896(2)

119894(+1199083Δ119896(3)

119894)

1199081+ 1199082(+1199083)

(25)

where Δ119896(119895)

119894is the stiffness reduction of the damaged story

119894th interpolated by the change in square frequency of the 119895th

International Scholarly Research Notices 5

mode (Δ(119895)

119894) and 119908

119895is a weight with the variance given by

119908119895= 1(120590

Δ(119895))2

The reliability band of Δ119896(119895)119894

is obtained from 120590Δ(119895)

119894

in (22)and the standard deviation of DQI (120590DQI ) is also calculated

4 Performance of Proposed Method

A four-story shear structure was measured to show thefeasibility of the proposed method It was modeled as a one-dimensional lumped mass shear model as shown in Figure 1The damping ratio for each mode was chosen to be 3The sampling frequency 200Hz is commonly used in majormonitoring systems for real buildings However as we canonly need lower natural frequencies the sampling frequencycan be as low as 50Hz provided that an appropriate antialias-ing filter is used In this study the data sampling frequencywas chosen 200Hz The excitation loading is white-noiseAlthough we used white-noise in this paper the excitationcan be any such as winds microtremors and earthquakes

The stories of the structure had the same mass 119898119894=

1000 tons but their stiffness was different 1198961= 119896 = 13 times

103MNm 119896

2= 09119896 119896

3= 08119896 and 119896

4= 07119896 The first two

natural frequencies of the undamaged structure were 189Hzand 519Hz Four cases of damage (damage to the 1st 2nd3rd and 4th stories) were studiedThe damage was simulatedby reducing the stiffness of each story by 5 10 15 2025 and 30

Figure 2 shows the sensitivity of the changes in the firsttwo natural frequencies to the location of the damaged storyFrom Figure 2 it can be seen that the influence of the damagelocation identified by the first natural frequency decreaseson higher stories the results mean that the first naturalfrequency is the most sensitive to damage on the 1st story Bycontrast the second natural frequency is the least sensitive todamage on the 2nd story

Figure 3 uses different colors to plot DLI12 values andtheir confidence interval of 5 to 30 stiffness reduction foreach storyTheDLI12 values of these cases are indicated by thebold lines in the middle of each color their reliability bandconsidering 2120590DLI is indicated by the two thin lines above andbelow each thick line (the standard deviation of the squarednatural frequencies is assumed to be 1)

From Figure 3 we can see the following

(i) The DLI12 values are stable at each damaged story forany level of damage the location of the damaged storycan be detected by examining the correlation betweenthe unknown data and the data of the story on whichdamage was detected

(ii) The variances of 2120590DLI depend on the stiffness reduc-tion when the reduction in stiffness is smaller thesevalues become larger So the extent of damage asindicated by the stiffness reduction must be bigenough to detect the location of damage without anymistakes

In this simulation two modes were enough to detect thedamaged story After determining the location of the damage

Damaged story1 2 3 4

0005

01015

02

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 2nd mode Δ(2)

(a)

1 2 3 40

00501

01502

Damaged story

51015

202530

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 1st mode Δ(1)

(b)

Figure 2 Sensitivity to changes in square natural frequencies (Δ(1)

and Δ(2)) of 4-story structure

0010203040506070809

111

Stiffness reduction ()

2nd story

5 10 15 20 25 30

DLI

12

valu

es an

d th

eir i

nter

vals

damage

1st storydamage3rd storydamage

4th storydamage

Figure 3 DLI12 values and their intervals within the range ⟨DLI plusmn2120590DLI⟩ of 4-story structure

by using the DLI values presented in Figure 3 the quan-tification of damage on that story can be calculated fromthe DQI definition with its standard deviation Figure 4compares the DQI with its uncertainty against real damageto a 4-story structure that reduced the stiffness of each storyby 5 15 and 25

5 Application to Tall Buildings

Next we verified ourmethod on an eight-story structureThedamping ratio was chosen to be 3 and the data samplingfrequency was 200Hz the excitation loading is white-noiseThemass of each story was 1000 tonsThe stiffness of the firststory was assumed to be 119896

1= 119896 = 13 times 10

3MNm and thestiffness of the other stories was 119896

2= 095119896 119896

3= 09119896 119896

4=

085119896 1198965= 08119896 119896

6= 075119896 119896

7= 07119896 and 119896

8= 065119896

6 International Scholarly Research Notices

05

1015202530

Stiff

ness

redu

ctio

n (

)

DQIReal damage

Damage to1st story

Damage to2nd story

Damage to3rd story

Damage to4th story

Figure 4 Comparison of DQI with uncertainty and actual damageof the 4-story structure

Table 1 Damage scenarios of eight-story structure

Damage casenumber

Location of damage(damaged story)

Quantification of damages(percentage of the stiffness

reduction)1 1st 232 4th 143 7th 6

The first three natural frequencies of the undamagedstructure were calculated to be 099 282 and 458Hz Table 1summarizes these three cases in this simulation The firstthree natural frequencies were obtained from accelerationdata of the sensor on top of the simulated structure byapplying the subspace identification method described in[19] The resulting DLI values are displayed in Table 2

Eight cases of damage (from the 1st to 8th story) weresimulated For each case the damage was simulated byreducing the stiffness of each story from5 to 25 and these 21levels of damage were reflected in the changes in the naturalfrequencies The DLI12 values and their reliability band werecalculated by assuming that the squared frequency had astandard deviation of 1 (Figure 5) There is some overlap ofthe 95 confidence intervals The DLI12 values could detectthe location of the damage in three sections of the structureThe DLI12 values of the three damage cases are plotted as thedash dotted lines in Figure 5

The next step is using the 2nd and 3rd frequency changesto get DLI23 and their reliability band Figures 6 7 and 8 plotthe DLI23 values

After determining the location of the damage by usingthe DLI values the amount of damage on that story wascalculated by interpolating the DQI valuesTheDLI and DQIvalues together with their uncertainty in the three damagecases are displayed as the red crosses in Figures 6 to 8

Damage indices are dependent on the mass and stiffnessdistribution Thus we need a prior knowledge of themto apply our method If design drawing is not availablesome system identification tools may be needed This is thelimitation of this method to be applied to a real building

0010203040506070809

111

Stiffness reduction ()5 10 15 20 25

DLI

12

valu

es an

d th

eir i

nter

vals

minus01

Case 2 (095)

3rd and 4th story

Case 1 (051)

1st 2nd and 5th

Case 3 (015)

6th 7th and 8th

damage

story damage

story damage

Figure 5 DLI12 values and their intervals for 8-story structure

040506070809

1

DQI-stiffness reduction ()

Case 1

5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

2nd story damage

1st story damage

5th story damage

Figure 6 DLI23 and DQI values and their intervals for the bluesections

6 Conclusion

The presented method determines the location and amountof damage to shear structures by using the changes in the firstfew natural frequencies only Natural frequencies decrease asa result of damage and the damage indices (DLI andDQI) usethese changes to reveal the location and amount of damageTheDLI uses twonatural frequency changes between undam-aged and damaged states and is stable for each damaged storyAfter identifying the location of the damage the DQI of thedamaged story is used to quantify the damage

As we need only a few natural frequencies two vibrationsensors are enough to obtain the modal frequencies one onthe ground detecting an input and the other on the roofdetecting an output If the input lasts long and the spectrumis flat we may identify those parameters using the outputdata without input informationThus in such a case only onesensor is needed

The uncertainty associated with system identificationmethods for obtaining natural frequencies was also carefullyconsidered and the confidence intervals of the DLI valueswere acceptable with high accuracy The method was alsoshown to be able to detect the location of damage to tallbuildings

Conflict of Interests

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

International Scholarly Research Notices 7

Table 2 Natural frequencies of 3 damage cases of eight-story structure

Damage casenumber

The 1st natural frequency 1205961

[Hz]The 2nd natural frequency 120596

2

[Hz]The 3rd natural frequency 120596

3

[Hz] DLI12 DLI23

1 097 275 447 051 0522 098 282 450 095 0043 099 276 449 015 055

00102030405

Case 2

DQI-stiffness reduction () 5 10 15 20 25minus01

3rd storydamage

4th storydamage

DLI

23

valu

es an

dth

eir i

nter

vals

Figure 7 DLI23 and DQI values and their intervals for the greensection

0203040506070809

111

Case 3

DQI-stiffness reduction () 5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

6th story damage

7th story damage

8th story damage

Figure 8 DLI23 and DQI values and their intervals for the purplesection

Acknowledgments

This work was supported in part by a Grant-in-Aid no22310103 (PI Akira Mita) and Grant-in-Aid to the GlobalCenter of Excellence Program for the ldquoCenter for Educationand Research of Symbiotic Safe and Secure System Designrdquofrom the Ministry of Education Culture Sport Science andTechnology of Japan

References

[1] A Mita ldquoEmerging needs in Japan for health monitoring tech-nologies in civil and building structuresrdquo in Proceedings of the2nd International Workshop on Structural Health Monitoringpp 56ndash67 Stanford University Stanford Calif USA September1999

[2] A Mita Structural Dynamic for Health Monitoring SankeishaNagoya Japan 2003

[3] S W Doebling C R Farrar M B Prime and D W ShevitzDamage Identification and Health Monitoring of Structural andMechanical Systems from Changes in Their Vibration Character-istics A Literature Review Los AlamosNational Laboratory LosAlamos NM USA 1996

[4] J Zhao and J T DeWolf ldquoSensitivity study for vibrationalparameters used in damage detectionrdquo Journal of StructuralEngineering vol 125 no 4 pp 410ndash416 1999

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

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

[7] A Messina I A John and E J Williams ldquoDamage detectionand localization using natural frequency changesrdquo in Proceed-ings of the Conference on Identification in Engineering Systemspp 67ndash76 Swansea UK 1996

[8] P E Carden and A Mita ldquoChallenges in developing confidenceintervals on modal parameters estimated for large civil infras-tructure with stochastic subspace identificationrdquo StructuralControl and Health Monitoring vol 18 no 1 pp 53ndash78 2011

[9] W Gao ldquoInterval natural frequency and mode shape analysisfor truss structures with interval parametersrdquo Finite Elements inAnalysis and Design vol 42 no 6 pp 471ndash477 2006

[10] W Gao ldquoNatural frequency and mode shape analysis ofstructures with uncertaintyrdquo Mechanical Systems and SignalProcessing vol 21 no 1 pp 24ndash39 2007

[11] P Moser and B Moaveni ldquoEnvironmental effects on theidentified natural frequencies of the Dowling Hall FootbridgerdquoMechanical Systems and Signal Processing vol 25 no 7 pp2336ndash2357 2011

[12] K He and W D Zhu ldquoStructural damage detection usingchanges in natural frequencies theory and applicationsrdquo Jour-nal of Physics Conference Series vol 305 no 1 Article ID012054 2011

[13] M D Vila F L Olivares and G R Alcorta ldquoUncertainties inthe estimation of natural frequencies of vibration in buildings ofMexico Cityrdquo Informacion Tecnologica vol 11 no 3 pp 177ndash1842000

[14] J F Clinton S C Bradford T H Heaton and J Favela ldquoTheobserved wander of the natural frequencies in a structurerdquoBulletin of the Seismological Society of America vol 96 no 1pp 237ndash257 2006

[15] J F Wang C C Lin and S M Yen ldquoA story damage indexof seismically-excited buildings based on modal frequency andmode shaperdquo Engineering Structures vol 29 no 9 pp 2143ndash2157 2007

[16] K Morita M Teshigawara and T Hamamoto ldquoDetection andestimation of damage to steel frames through shaking tabletestsrdquo Structural Control and Health Monitoring vol 12 no 3-4pp 357ndash380 2005

[17] V Lindberg Uncertainties and Error Propagation 2008 httpwwwriteducosuphysicsuncertaintiesUncertaintiespart2html

8 International Scholarly Research Notices

[18] H HoThu and A Mita ldquoApplicability of mode-based damageassessment methods to severely damaged steel buildingrdquo inSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 8345 of Proceedings of SPIE SanDiego Calif USA March 2012

[19] M Verhaegen and P Dewilde ldquoSubspace model identificationpart 1 the output-error state-space model identification class ofalgorithmsrdquo International Journal of Control vol 56 no 5 pp1187ndash1210 1992

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 Scholarly Research Notices 5

mode (Δ(119895)

119894) and 119908

119895is a weight with the variance given by

119908119895= 1(120590

Δ(119895))2

The reliability band of Δ119896(119895)119894

is obtained from 120590Δ(119895)

119894

in (22)and the standard deviation of DQI (120590DQI ) is also calculated

4 Performance of Proposed Method

A four-story shear structure was measured to show thefeasibility of the proposed method It was modeled as a one-dimensional lumped mass shear model as shown in Figure 1The damping ratio for each mode was chosen to be 3The sampling frequency 200Hz is commonly used in majormonitoring systems for real buildings However as we canonly need lower natural frequencies the sampling frequencycan be as low as 50Hz provided that an appropriate antialias-ing filter is used In this study the data sampling frequencywas chosen 200Hz The excitation loading is white-noiseAlthough we used white-noise in this paper the excitationcan be any such as winds microtremors and earthquakes

The stories of the structure had the same mass 119898119894=

1000 tons but their stiffness was different 1198961= 119896 = 13 times

103MNm 119896

2= 09119896 119896

3= 08119896 and 119896

4= 07119896 The first two

natural frequencies of the undamaged structure were 189Hzand 519Hz Four cases of damage (damage to the 1st 2nd3rd and 4th stories) were studiedThe damage was simulatedby reducing the stiffness of each story by 5 10 15 2025 and 30

Figure 2 shows the sensitivity of the changes in the firsttwo natural frequencies to the location of the damaged storyFrom Figure 2 it can be seen that the influence of the damagelocation identified by the first natural frequency decreaseson higher stories the results mean that the first naturalfrequency is the most sensitive to damage on the 1st story Bycontrast the second natural frequency is the least sensitive todamage on the 2nd story

Figure 3 uses different colors to plot DLI12 values andtheir confidence interval of 5 to 30 stiffness reduction foreach storyTheDLI12 values of these cases are indicated by thebold lines in the middle of each color their reliability bandconsidering 2120590DLI is indicated by the two thin lines above andbelow each thick line (the standard deviation of the squarednatural frequencies is assumed to be 1)

From Figure 3 we can see the following

(i) The DLI12 values are stable at each damaged story forany level of damage the location of the damaged storycan be detected by examining the correlation betweenthe unknown data and the data of the story on whichdamage was detected

(ii) The variances of 2120590DLI depend on the stiffness reduc-tion when the reduction in stiffness is smaller thesevalues become larger So the extent of damage asindicated by the stiffness reduction must be bigenough to detect the location of damage without anymistakes

In this simulation two modes were enough to detect thedamaged story After determining the location of the damage

Damaged story1 2 3 4

0005

01015

02

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 2nd mode Δ(2)

(a)

1 2 3 40

00501

01502

Damaged story

51015

202530

Sens

itivi

ty o

f squ

ared

natu

ral f

requ

enci

es (H

z)

Change in squared frequency of 1st mode Δ(1)

(b)

Figure 2 Sensitivity to changes in square natural frequencies (Δ(1)

and Δ(2)) of 4-story structure

0010203040506070809

111

Stiffness reduction ()

2nd story

5 10 15 20 25 30

DLI

12

valu

es an

d th

eir i

nter

vals

damage

1st storydamage3rd storydamage

4th storydamage

Figure 3 DLI12 values and their intervals within the range ⟨DLI plusmn2120590DLI⟩ of 4-story structure

by using the DLI values presented in Figure 3 the quan-tification of damage on that story can be calculated fromthe DQI definition with its standard deviation Figure 4compares the DQI with its uncertainty against real damageto a 4-story structure that reduced the stiffness of each storyby 5 15 and 25

5 Application to Tall Buildings

Next we verified ourmethod on an eight-story structureThedamping ratio was chosen to be 3 and the data samplingfrequency was 200Hz the excitation loading is white-noiseThemass of each story was 1000 tonsThe stiffness of the firststory was assumed to be 119896

1= 119896 = 13 times 10

3MNm and thestiffness of the other stories was 119896

2= 095119896 119896

3= 09119896 119896

4=

085119896 1198965= 08119896 119896

6= 075119896 119896

7= 07119896 and 119896

8= 065119896

6 International Scholarly Research Notices

05

1015202530

Stiff

ness

redu

ctio

n (

)

DQIReal damage

Damage to1st story

Damage to2nd story

Damage to3rd story

Damage to4th story

Figure 4 Comparison of DQI with uncertainty and actual damageof the 4-story structure

Table 1 Damage scenarios of eight-story structure

Damage casenumber

Location of damage(damaged story)

Quantification of damages(percentage of the stiffness

reduction)1 1st 232 4th 143 7th 6

The first three natural frequencies of the undamagedstructure were calculated to be 099 282 and 458Hz Table 1summarizes these three cases in this simulation The firstthree natural frequencies were obtained from accelerationdata of the sensor on top of the simulated structure byapplying the subspace identification method described in[19] The resulting DLI values are displayed in Table 2

Eight cases of damage (from the 1st to 8th story) weresimulated For each case the damage was simulated byreducing the stiffness of each story from5 to 25 and these 21levels of damage were reflected in the changes in the naturalfrequencies The DLI12 values and their reliability band werecalculated by assuming that the squared frequency had astandard deviation of 1 (Figure 5) There is some overlap ofthe 95 confidence intervals The DLI12 values could detectthe location of the damage in three sections of the structureThe DLI12 values of the three damage cases are plotted as thedash dotted lines in Figure 5

The next step is using the 2nd and 3rd frequency changesto get DLI23 and their reliability band Figures 6 7 and 8 plotthe DLI23 values

After determining the location of the damage by usingthe DLI values the amount of damage on that story wascalculated by interpolating the DQI valuesTheDLI and DQIvalues together with their uncertainty in the three damagecases are displayed as the red crosses in Figures 6 to 8

Damage indices are dependent on the mass and stiffnessdistribution Thus we need a prior knowledge of themto apply our method If design drawing is not availablesome system identification tools may be needed This is thelimitation of this method to be applied to a real building

0010203040506070809

111

Stiffness reduction ()5 10 15 20 25

DLI

12

valu

es an

d th

eir i

nter

vals

minus01

Case 2 (095)

3rd and 4th story

Case 1 (051)

1st 2nd and 5th

Case 3 (015)

6th 7th and 8th

damage

story damage

story damage

Figure 5 DLI12 values and their intervals for 8-story structure

040506070809

1

DQI-stiffness reduction ()

Case 1

5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

2nd story damage

1st story damage

5th story damage

Figure 6 DLI23 and DQI values and their intervals for the bluesections

6 Conclusion

The presented method determines the location and amountof damage to shear structures by using the changes in the firstfew natural frequencies only Natural frequencies decrease asa result of damage and the damage indices (DLI andDQI) usethese changes to reveal the location and amount of damageTheDLI uses twonatural frequency changes between undam-aged and damaged states and is stable for each damaged storyAfter identifying the location of the damage the DQI of thedamaged story is used to quantify the damage

As we need only a few natural frequencies two vibrationsensors are enough to obtain the modal frequencies one onthe ground detecting an input and the other on the roofdetecting an output If the input lasts long and the spectrumis flat we may identify those parameters using the outputdata without input informationThus in such a case only onesensor is needed

The uncertainty associated with system identificationmethods for obtaining natural frequencies was also carefullyconsidered and the confidence intervals of the DLI valueswere acceptable with high accuracy The method was alsoshown to be able to detect the location of damage to tallbuildings

Conflict of Interests

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

International Scholarly Research Notices 7

Table 2 Natural frequencies of 3 damage cases of eight-story structure

Damage casenumber

The 1st natural frequency 1205961

[Hz]The 2nd natural frequency 120596

2

[Hz]The 3rd natural frequency 120596

3

[Hz] DLI12 DLI23

1 097 275 447 051 0522 098 282 450 095 0043 099 276 449 015 055

00102030405

Case 2

DQI-stiffness reduction () 5 10 15 20 25minus01

3rd storydamage

4th storydamage

DLI

23

valu

es an

dth

eir i

nter

vals

Figure 7 DLI23 and DQI values and their intervals for the greensection

0203040506070809

111

Case 3

DQI-stiffness reduction () 5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

6th story damage

7th story damage

8th story damage

Figure 8 DLI23 and DQI values and their intervals for the purplesection

Acknowledgments

This work was supported in part by a Grant-in-Aid no22310103 (PI Akira Mita) and Grant-in-Aid to the GlobalCenter of Excellence Program for the ldquoCenter for Educationand Research of Symbiotic Safe and Secure System Designrdquofrom the Ministry of Education Culture Sport Science andTechnology of Japan

References

[1] A Mita ldquoEmerging needs in Japan for health monitoring tech-nologies in civil and building structuresrdquo in Proceedings of the2nd International Workshop on Structural Health Monitoringpp 56ndash67 Stanford University Stanford Calif USA September1999

[2] A Mita Structural Dynamic for Health Monitoring SankeishaNagoya Japan 2003

[3] S W Doebling C R Farrar M B Prime and D W ShevitzDamage Identification and Health Monitoring of Structural andMechanical Systems from Changes in Their Vibration Character-istics A Literature Review Los AlamosNational Laboratory LosAlamos NM USA 1996

[4] J Zhao and J T DeWolf ldquoSensitivity study for vibrationalparameters used in damage detectionrdquo Journal of StructuralEngineering vol 125 no 4 pp 410ndash416 1999

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

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

[7] A Messina I A John and E J Williams ldquoDamage detectionand localization using natural frequency changesrdquo in Proceed-ings of the Conference on Identification in Engineering Systemspp 67ndash76 Swansea UK 1996

[8] P E Carden and A Mita ldquoChallenges in developing confidenceintervals on modal parameters estimated for large civil infras-tructure with stochastic subspace identificationrdquo StructuralControl and Health Monitoring vol 18 no 1 pp 53ndash78 2011

[9] W Gao ldquoInterval natural frequency and mode shape analysisfor truss structures with interval parametersrdquo Finite Elements inAnalysis and Design vol 42 no 6 pp 471ndash477 2006

[10] W Gao ldquoNatural frequency and mode shape analysis ofstructures with uncertaintyrdquo Mechanical Systems and SignalProcessing vol 21 no 1 pp 24ndash39 2007

[11] P Moser and B Moaveni ldquoEnvironmental effects on theidentified natural frequencies of the Dowling Hall FootbridgerdquoMechanical Systems and Signal Processing vol 25 no 7 pp2336ndash2357 2011

[12] K He and W D Zhu ldquoStructural damage detection usingchanges in natural frequencies theory and applicationsrdquo Jour-nal of Physics Conference Series vol 305 no 1 Article ID012054 2011

[13] M D Vila F L Olivares and G R Alcorta ldquoUncertainties inthe estimation of natural frequencies of vibration in buildings ofMexico Cityrdquo Informacion Tecnologica vol 11 no 3 pp 177ndash1842000

[14] J F Clinton S C Bradford T H Heaton and J Favela ldquoTheobserved wander of the natural frequencies in a structurerdquoBulletin of the Seismological Society of America vol 96 no 1pp 237ndash257 2006

[15] J F Wang C C Lin and S M Yen ldquoA story damage indexof seismically-excited buildings based on modal frequency andmode shaperdquo Engineering Structures vol 29 no 9 pp 2143ndash2157 2007

[16] K Morita M Teshigawara and T Hamamoto ldquoDetection andestimation of damage to steel frames through shaking tabletestsrdquo Structural Control and Health Monitoring vol 12 no 3-4pp 357ndash380 2005

[17] V Lindberg Uncertainties and Error Propagation 2008 httpwwwriteducosuphysicsuncertaintiesUncertaintiespart2html

8 International Scholarly Research Notices

[18] H HoThu and A Mita ldquoApplicability of mode-based damageassessment methods to severely damaged steel buildingrdquo inSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 8345 of Proceedings of SPIE SanDiego Calif USA March 2012

[19] M Verhaegen and P Dewilde ldquoSubspace model identificationpart 1 the output-error state-space model identification class ofalgorithmsrdquo International Journal of Control vol 56 no 5 pp1187ndash1210 1992

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 International Scholarly Research Notices

05

1015202530

Stiff

ness

redu

ctio

n (

)

DQIReal damage

Damage to1st story

Damage to2nd story

Damage to3rd story

Damage to4th story

Figure 4 Comparison of DQI with uncertainty and actual damageof the 4-story structure

Table 1 Damage scenarios of eight-story structure

Damage casenumber

Location of damage(damaged story)

Quantification of damages(percentage of the stiffness

reduction)1 1st 232 4th 143 7th 6

The first three natural frequencies of the undamagedstructure were calculated to be 099 282 and 458Hz Table 1summarizes these three cases in this simulation The firstthree natural frequencies were obtained from accelerationdata of the sensor on top of the simulated structure byapplying the subspace identification method described in[19] The resulting DLI values are displayed in Table 2

Eight cases of damage (from the 1st to 8th story) weresimulated For each case the damage was simulated byreducing the stiffness of each story from5 to 25 and these 21levels of damage were reflected in the changes in the naturalfrequencies The DLI12 values and their reliability band werecalculated by assuming that the squared frequency had astandard deviation of 1 (Figure 5) There is some overlap ofthe 95 confidence intervals The DLI12 values could detectthe location of the damage in three sections of the structureThe DLI12 values of the three damage cases are plotted as thedash dotted lines in Figure 5

The next step is using the 2nd and 3rd frequency changesto get DLI23 and their reliability band Figures 6 7 and 8 plotthe DLI23 values

After determining the location of the damage by usingthe DLI values the amount of damage on that story wascalculated by interpolating the DQI valuesTheDLI and DQIvalues together with their uncertainty in the three damagecases are displayed as the red crosses in Figures 6 to 8

Damage indices are dependent on the mass and stiffnessdistribution Thus we need a prior knowledge of themto apply our method If design drawing is not availablesome system identification tools may be needed This is thelimitation of this method to be applied to a real building

0010203040506070809

111

Stiffness reduction ()5 10 15 20 25

DLI

12

valu

es an

d th

eir i

nter

vals

minus01

Case 2 (095)

3rd and 4th story

Case 1 (051)

1st 2nd and 5th

Case 3 (015)

6th 7th and 8th

damage

story damage

story damage

Figure 5 DLI12 values and their intervals for 8-story structure

040506070809

1

DQI-stiffness reduction ()

Case 1

5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

2nd story damage

1st story damage

5th story damage

Figure 6 DLI23 and DQI values and their intervals for the bluesections

6 Conclusion

The presented method determines the location and amountof damage to shear structures by using the changes in the firstfew natural frequencies only Natural frequencies decrease asa result of damage and the damage indices (DLI andDQI) usethese changes to reveal the location and amount of damageTheDLI uses twonatural frequency changes between undam-aged and damaged states and is stable for each damaged storyAfter identifying the location of the damage the DQI of thedamaged story is used to quantify the damage

As we need only a few natural frequencies two vibrationsensors are enough to obtain the modal frequencies one onthe ground detecting an input and the other on the roofdetecting an output If the input lasts long and the spectrumis flat we may identify those parameters using the outputdata without input informationThus in such a case only onesensor is needed

The uncertainty associated with system identificationmethods for obtaining natural frequencies was also carefullyconsidered and the confidence intervals of the DLI valueswere acceptable with high accuracy The method was alsoshown to be able to detect the location of damage to tallbuildings

Conflict of Interests

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

International Scholarly Research Notices 7

Table 2 Natural frequencies of 3 damage cases of eight-story structure

Damage casenumber

The 1st natural frequency 1205961

[Hz]The 2nd natural frequency 120596

2

[Hz]The 3rd natural frequency 120596

3

[Hz] DLI12 DLI23

1 097 275 447 051 0522 098 282 450 095 0043 099 276 449 015 055

00102030405

Case 2

DQI-stiffness reduction () 5 10 15 20 25minus01

3rd storydamage

4th storydamage

DLI

23

valu

es an

dth

eir i

nter

vals

Figure 7 DLI23 and DQI values and their intervals for the greensection

0203040506070809

111

Case 3

DQI-stiffness reduction () 5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

6th story damage

7th story damage

8th story damage

Figure 8 DLI23 and DQI values and their intervals for the purplesection

Acknowledgments

This work was supported in part by a Grant-in-Aid no22310103 (PI Akira Mita) and Grant-in-Aid to the GlobalCenter of Excellence Program for the ldquoCenter for Educationand Research of Symbiotic Safe and Secure System Designrdquofrom the Ministry of Education Culture Sport Science andTechnology of Japan

References

[1] A Mita ldquoEmerging needs in Japan for health monitoring tech-nologies in civil and building structuresrdquo in Proceedings of the2nd International Workshop on Structural Health Monitoringpp 56ndash67 Stanford University Stanford Calif USA September1999

[2] A Mita Structural Dynamic for Health Monitoring SankeishaNagoya Japan 2003

[3] S W Doebling C R Farrar M B Prime and D W ShevitzDamage Identification and Health Monitoring of Structural andMechanical Systems from Changes in Their Vibration Character-istics A Literature Review Los AlamosNational Laboratory LosAlamos NM USA 1996

[4] J Zhao and J T DeWolf ldquoSensitivity study for vibrationalparameters used in damage detectionrdquo Journal of StructuralEngineering vol 125 no 4 pp 410ndash416 1999

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

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

[7] A Messina I A John and E J Williams ldquoDamage detectionand localization using natural frequency changesrdquo in Proceed-ings of the Conference on Identification in Engineering Systemspp 67ndash76 Swansea UK 1996

[8] P E Carden and A Mita ldquoChallenges in developing confidenceintervals on modal parameters estimated for large civil infras-tructure with stochastic subspace identificationrdquo StructuralControl and Health Monitoring vol 18 no 1 pp 53ndash78 2011

[9] W Gao ldquoInterval natural frequency and mode shape analysisfor truss structures with interval parametersrdquo Finite Elements inAnalysis and Design vol 42 no 6 pp 471ndash477 2006

[10] W Gao ldquoNatural frequency and mode shape analysis ofstructures with uncertaintyrdquo Mechanical Systems and SignalProcessing vol 21 no 1 pp 24ndash39 2007

[11] P Moser and B Moaveni ldquoEnvironmental effects on theidentified natural frequencies of the Dowling Hall FootbridgerdquoMechanical Systems and Signal Processing vol 25 no 7 pp2336ndash2357 2011

[12] K He and W D Zhu ldquoStructural damage detection usingchanges in natural frequencies theory and applicationsrdquo Jour-nal of Physics Conference Series vol 305 no 1 Article ID012054 2011

[13] M D Vila F L Olivares and G R Alcorta ldquoUncertainties inthe estimation of natural frequencies of vibration in buildings ofMexico Cityrdquo Informacion Tecnologica vol 11 no 3 pp 177ndash1842000

[14] J F Clinton S C Bradford T H Heaton and J Favela ldquoTheobserved wander of the natural frequencies in a structurerdquoBulletin of the Seismological Society of America vol 96 no 1pp 237ndash257 2006

[15] J F Wang C C Lin and S M Yen ldquoA story damage indexof seismically-excited buildings based on modal frequency andmode shaperdquo Engineering Structures vol 29 no 9 pp 2143ndash2157 2007

[16] K Morita M Teshigawara and T Hamamoto ldquoDetection andestimation of damage to steel frames through shaking tabletestsrdquo Structural Control and Health Monitoring vol 12 no 3-4pp 357ndash380 2005

[17] V Lindberg Uncertainties and Error Propagation 2008 httpwwwriteducosuphysicsuncertaintiesUncertaintiespart2html

8 International Scholarly Research Notices

[18] H HoThu and A Mita ldquoApplicability of mode-based damageassessment methods to severely damaged steel buildingrdquo inSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 8345 of Proceedings of SPIE SanDiego Calif USA March 2012

[19] M Verhaegen and P Dewilde ldquoSubspace model identificationpart 1 the output-error state-space model identification class ofalgorithmsrdquo International Journal of Control vol 56 no 5 pp1187ndash1210 1992

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 Scholarly Research Notices 7

Table 2 Natural frequencies of 3 damage cases of eight-story structure

Damage casenumber

The 1st natural frequency 1205961

[Hz]The 2nd natural frequency 120596

2

[Hz]The 3rd natural frequency 120596

3

[Hz] DLI12 DLI23

1 097 275 447 051 0522 098 282 450 095 0043 099 276 449 015 055

00102030405

Case 2

DQI-stiffness reduction () 5 10 15 20 25minus01

3rd storydamage

4th storydamage

DLI

23

valu

es an

dth

eir i

nter

vals

Figure 7 DLI23 and DQI values and their intervals for the greensection

0203040506070809

111

Case 3

DQI-stiffness reduction () 5 10 15 20 25

DLI

23

valu

es an

d th

eir i

nter

vals

6th story damage

7th story damage

8th story damage

Figure 8 DLI23 and DQI values and their intervals for the purplesection

Acknowledgments

This work was supported in part by a Grant-in-Aid no22310103 (PI Akira Mita) and Grant-in-Aid to the GlobalCenter of Excellence Program for the ldquoCenter for Educationand Research of Symbiotic Safe and Secure System Designrdquofrom the Ministry of Education Culture Sport Science andTechnology of Japan

References

[1] A Mita ldquoEmerging needs in Japan for health monitoring tech-nologies in civil and building structuresrdquo in Proceedings of the2nd International Workshop on Structural Health Monitoringpp 56ndash67 Stanford University Stanford Calif USA September1999

[2] A Mita Structural Dynamic for Health Monitoring SankeishaNagoya Japan 2003

[3] S W Doebling C R Farrar M B Prime and D W ShevitzDamage Identification and Health Monitoring of Structural andMechanical Systems from Changes in Their Vibration Character-istics A Literature Review Los AlamosNational Laboratory LosAlamos NM USA 1996

[4] J Zhao and J T DeWolf ldquoSensitivity study for vibrationalparameters used in damage detectionrdquo Journal of StructuralEngineering vol 125 no 4 pp 410ndash416 1999

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

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

[7] A Messina I A John and E J Williams ldquoDamage detectionand localization using natural frequency changesrdquo in Proceed-ings of the Conference on Identification in Engineering Systemspp 67ndash76 Swansea UK 1996

[8] P E Carden and A Mita ldquoChallenges in developing confidenceintervals on modal parameters estimated for large civil infras-tructure with stochastic subspace identificationrdquo StructuralControl and Health Monitoring vol 18 no 1 pp 53ndash78 2011

[9] W Gao ldquoInterval natural frequency and mode shape analysisfor truss structures with interval parametersrdquo Finite Elements inAnalysis and Design vol 42 no 6 pp 471ndash477 2006

[10] W Gao ldquoNatural frequency and mode shape analysis ofstructures with uncertaintyrdquo Mechanical Systems and SignalProcessing vol 21 no 1 pp 24ndash39 2007

[11] P Moser and B Moaveni ldquoEnvironmental effects on theidentified natural frequencies of the Dowling Hall FootbridgerdquoMechanical Systems and Signal Processing vol 25 no 7 pp2336ndash2357 2011

[12] K He and W D Zhu ldquoStructural damage detection usingchanges in natural frequencies theory and applicationsrdquo Jour-nal of Physics Conference Series vol 305 no 1 Article ID012054 2011

[13] M D Vila F L Olivares and G R Alcorta ldquoUncertainties inthe estimation of natural frequencies of vibration in buildings ofMexico Cityrdquo Informacion Tecnologica vol 11 no 3 pp 177ndash1842000

[14] J F Clinton S C Bradford T H Heaton and J Favela ldquoTheobserved wander of the natural frequencies in a structurerdquoBulletin of the Seismological Society of America vol 96 no 1pp 237ndash257 2006

[15] J F Wang C C Lin and S M Yen ldquoA story damage indexof seismically-excited buildings based on modal frequency andmode shaperdquo Engineering Structures vol 29 no 9 pp 2143ndash2157 2007

[16] K Morita M Teshigawara and T Hamamoto ldquoDetection andestimation of damage to steel frames through shaking tabletestsrdquo Structural Control and Health Monitoring vol 12 no 3-4pp 357ndash380 2005

[17] V Lindberg Uncertainties and Error Propagation 2008 httpwwwriteducosuphysicsuncertaintiesUncertaintiespart2html

8 International Scholarly Research Notices

[18] H HoThu and A Mita ldquoApplicability of mode-based damageassessment methods to severely damaged steel buildingrdquo inSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 8345 of Proceedings of SPIE SanDiego Calif USA March 2012

[19] M Verhaegen and P Dewilde ldquoSubspace model identificationpart 1 the output-error state-space model identification class ofalgorithmsrdquo International Journal of Control vol 56 no 5 pp1187ndash1210 1992

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 International Scholarly Research Notices

[18] H HoThu and A Mita ldquoApplicability of mode-based damageassessment methods to severely damaged steel buildingrdquo inSensors and Smart Structures Technologies for Civil Mechanicaland Aerospace Systems vol 8345 of Proceedings of SPIE SanDiego Calif USA March 2012

[19] M Verhaegen and P Dewilde ldquoSubspace model identificationpart 1 the output-error state-space model identification class ofalgorithmsrdquo International Journal of Control vol 56 no 5 pp1187ndash1210 1992

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

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

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