A Probabilistic Damage Identification Approach for ...In this paper, a probabilistic approach is proposed for the identification of structural damage and unknown external excitations

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013, Article ID 759102, 7 pageshttp://dx.doi.org/10.1155/2013/759102

Research ArticleA Probabilistic Damage Identification Approach for Structuresunder Unknown Excitation and with Measurement Uncertainties

Ying Lei,1 Ying Su,1 and Wenai Shen2

1 Department of Civil Engineering, Xiamen University, Xiamen 361005, China2Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Correspondence should be addressed to Ying Lei; [email protected]

Received 26 December 2012; Revised 4 May 2013; Accepted 26 May 2013

Academic Editor: Xiaojun Wang

Copyright © 2013 Ying Lei et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recently, an innovative algorithm has been proposed by the authors for the identification of structural damage under unknownexternal excitations. However, identification accuracy of this proposed deterministic algorithm decreases under high levelof measurement noise. A probabilistic approach is therefore proposed in this paper for damage identification consideringmeasurement noise uncertainties. Based on the former deterministic algorithm, the statistical values of the identified structuralparameters are estimated using the statistical theory and a damage index is defined.The probability of identified structural damageis further derived based on the reliability theory. The unknown external excitations to the structure are also identified by statisticalevaluation. A numerical example of the identification of structural damage of a multistory shear-type building and its unknownexcitation shows that the proposed probabilistic approach can accurately identify structural damage and the unknown excitationsusing only partial measurements of structural acceleration responses contaminated by intensive measurement noises.

1. Introduction

Structural damage detection is an important task for struc-tural health monitoring [1–5]. Usually, it is straightforwardto identify structural damage based on tracking the changesof the identified values of structural physical parameters, forexample, the degrading of element stiffness parameters. Inpractice, it is often impossible to deploy so many sensors thataccurately measure all excitation inputs and response outputsof systems. It is highly desirable to deploy as few sensorsas possible, so it is essential to explore efficient algorithmswhich can identify structural damage utilizing only a limitednumber of measured responses of structures subject to someunknown (unmeasured) excitations.

In the past decades, some researchers have proposed algo-rithms for simultaneous identification of structural parame-ters and unknown excitation, for example, the iterative least-square estimation approach [6, 7], the statistical averagealgorithm [8], the recursive least-square estimation [9],genetic algorithms [10], hybrid identification method [11],the dynamic response sensitivity method [11], the extendedKalman filter with unknown excitation inputs (EKF-UI) [12],

the sequential nonlinear least-square estimation (SNLSE)[13], and structural parameters and dynamic loading identifi-cation from incomplete measurements [14]. However, theseapproaches suffered the deficiencies of either all structuralresponse being assumed available or the analytical andnumerical identification procedures being rather complex.

Recently, an innovative algorithm has been proposed bythe authors for the identification of structural damage underunknown excitations using limited measurements of struc-tural acceleration responses [15, 16]. The proposed algorithmis based on the sequential utilization of the extended Kalmanestimator [17] for the recursive estimation of the extendedstate vector of a structure and the least-square estimationof its unknown excitation; that is, recursive solution forextended state vector is initially estimated followed by thesubsequent estimation of the unknown excitation via least-square estimation. Thus, proposed algorithm simplifies theidentification problem compared with previous simultane-ous identification approaches [18, 19]. Structural damage isdetected from the changes of structural parameters at theelement level, such as the degradation of identified elementstiffness parameters. Such a straightforward derivation and

2 Journal of Applied Mathematics

analytical solution are not available in the previous liter-ature. However, former numerical examples indicated thatthe identification accuracy of this proposed deterministicalgorithm decreased with the increase of the measurementnoise level [15, 16]. Therefore, it is necessary to develop anapproach which can avoid the false identification of damagesin the deterministic identification algorithm induced by therelatively high level of measurement noise.

Since the inevitable measurement noises are intrinsicallyuncertain, the identification of structural parameter andexternal excitation using measurements with intensive mea-surement noises is essentially an uncertain problem [20, 21].The identification performed by deterministic methods oftenleads to incorrect identification results of structural damagesand a disagreement between the identified unknown excita-tion and its true value while consideration of uncertaintieshas received more and more attention in recent years [22–26]. In this paper, a probabilistic approach is proposed for theidentification of structural damage under unknown externalexcitations and with measurement noise uncertainties. Basedon the deterministic algorithm, the statistical values of theidentified structural parameters are estimated, and the prob-ability of identified structural damage is further derived usingthe statistical theory and probability method. The rest of thepaper is organized as follows. Section 2 briefly introducesthe former deterministic algorithm for the identificationof structural damage under unknown external excitations,Section 3 presents the proposed probabilistic identificationapproach based on the improvement of the determinis-tic algorithm using the statistical and probability theory,Section 4 shows a numerical example of the identification ofstructural damage of a multistory shear-type frame buildingand its unknown excitation to demonstrate the proposedprobabilistic approach, and Section 5 gives the conclusions ofthe paper.

2. Brief Introduction of the DeterministicAlgorithm for Identification of StructuralDamage under Unknown Excitations

The equations of motion of a linear structural system subjectto unknown external excitation can be written as

Mx (𝑡) + Cx (𝑡) + Kx (𝑡) = B𝑢f𝑢(𝑡) , (1)

in which x(𝑡), x(𝑡), and x(𝑡) are the vectors of displacement,velocity, and acceleration response, respectively; M, C, andK are the mass, damping, and stiffness matrices, respectively;f𝑢(𝑡) is an unmeasured external excitation vector; and B𝑢 isthe influence matrix associated with f𝑢(𝑡). Usually, the massof a structural system can be estimated with accuracy basedon its geometry and material information.

2.1. Estimation of the Extended State Vector. The extendedstate vector of the system is defined as

X = [X𝑇1,X𝑇2, 𝜃𝑇

]

𝑇

; X1= x;

X2= x; 𝜃

𝑇

= [𝜃

1, 𝜃

2, . . . , 𝜃

𝑛]

𝑇

,

(2)

where 𝜃𝑇 is a vector of the 𝑛-unknown structural parameters,such as damping and stiffness parameters. As the structuralparameters are constant, (1) can be written in the followinggeneral nonlinear differential state equations [15, 16]:

X = g (X, f𝑢) . (3)

Usually, only a limited number of accelerometers aredeployed in structures to measure acceleration responses.Therefore, the discretized observation equation can beexpressed as

y [𝑘] = h (X [𝑘]) + G𝑢f𝑢 [𝑘] + v [𝑘] , (4)

where G𝑢 = DM−1B𝑢; h(X[𝑘]) = DM−1{−(C𝜃)X2[𝑘] − (K)𝜃X1[𝑘]} in which y[𝑘] is observation vector (measured accel-

eration responses) at time 𝑡 = 𝑘 × Δ𝑡 with Δ𝑡 being thesampling time step, (C)𝜃 represents elements in the dampingmatrix C composed by the unknown parameters of dampingin the parametric vector 𝜃, (K)𝜃 represents the constitution ofstiffness matrixK analogously, f𝑢[𝑘],X

1[𝑘] and X

2[𝑘] are the

corresponding discretized values at time 𝑡 = 𝑘 × Δ𝑡, D is thematrix associated with the locations of accelerometers, andk[𝑘] is the measured noise vector assumed to be a Gaussianwhite noise vector with zero mean and a covariance matrixE[v𝑖v𝑇𝑗] = R𝑖𝑗𝛿

𝑖𝑗, where 𝛿ij is the Kronecker delta.

Based on the extended Kalman estimator [15, 16], theextended state vector at time 𝑡 = (𝑘+1)×Δ𝑡 can be estimatedwith the observation of (y[1], y[2], . . . , y[𝑘]) as follows:

X [𝑘 + 1 | 𝑘] = X [𝑘 + 1 | 𝑘] + K [𝑘]

× {y [𝑘] − ℎ (X [𝑘 | 𝑘 − 1] , f [𝑘])

−G𝑢f𝑢 [𝑘 | 𝑘]} ,

(5)

in which

X [𝑘 + 1 | 𝑘] = X [𝑘 | 𝑘 − 1] + ∫𝑡[𝑘+1]

𝑡[𝑘]

g (X, f𝑢) 𝑑𝑡, (6)

where X[𝑘+1 | 𝑘] and f𝑢[𝑘 | 𝑘] are the estimation ofX[𝑘+1]and f𝑢[𝑘] given (y[1], y[2], . . . , y[𝑘]), respectively, andK[𝑘]is the Kalman gain matrix [15, 16].

However, since the external excitation f𝑢(𝑡) is unknown, itis impossible to obtain the recursive solution for the extendedstate vector by the classical extended Kalman estimator alone.

2.2. Identification of the Unknown Excitations. Under theconditions: (i) the number of output measurements is greaterthan that of the unknown excitations and (ii) measurements(sensors) are available at all DOFs where the unknownexcitation f𝑢(𝑡) acts; that is, matrix G𝑢 in (4) is nonzero; theunknown excitations at time 𝑡 = (𝑘+1)×Δ𝑡 can be estimatedfrom (4) by the least-square estimation as [15, 16]

f𝑢 [𝑘 + 1 | 𝑘 + 1] = [(G𝑢)𝑇G𝑢]−1

(G𝑢)𝑇

× {y [𝑘 + 1] − h (X [𝑘 + 1 | 𝑘])} ,(7)

Journal of Applied Mathematics 3

in which f𝑢[𝑘 + 1 | 𝑘 + 1] is the estimation of f𝑢[𝑘 + 1] giventhe observation of (y[1], y[2], . . . , y[𝑘 + 1]).

Therefore, the proposed algorithm can identify structuralparameters and unknown excitation in a sequential manner,which simplifies the identification problem compared withother simultaneous identification work. Structural damageis detected from the changes of structural parameters at theelement level, such as the degradation of identified elementstiffness parameters. Such a straightforward derivation andanalytical solution are not available in the previous literature[15, 16].

However, former numerical examples indicated that theidentification accuracy of this proposed deterministic algo-rithm decreases under high level of measurements noise [15,16]. The identification performed by using the deterministicalgorithm leads to incorrect identification results of struc-tural damages and a disagreement between the identifiedexcitation and its true value. Consequently, it is necessary todevelop an approach for identifying the structural damageand unknown excitation when the measurements are con-taminated by intensive measurement noises.

3. A Probabilistic Approach forthe Identification of Structural Damagewith Intensive Measurement Noises

Since the inevitable measurement noises are intrinsicallyuncertain, identification of structural parameter and un-known excitation using measurements with intensive mea-surement noises is essentially an uncertain problem. Aprobabilistic approach is proposed herein based on thedeterministic algorithm described in Section 2.

3.1. The Statistical Results of Identification Values. In theobserved equation, (4), themeasured noise vector is assumedto be a Gaussian white noise vector; that is, uncertainties inthe measured responses are assumed as normally distributedrandom variables. Then the measured acceleration responsevector y is an observation vector with uncertainties. Inpractice,many sets ofmeasured accelerations can be obtainedby repetitious experiments or long-term measurement ofstructures. In the numerical simulation, many sets of mea-sured accelerations can be obtained by the theoreticallycomputed responses superimposed with many sets of mea-surement noise with uncertainties. Then, each set of themeasured accelerations is used as an observation vector toidentify the structural parameters and unknown excitation byusing the deterministic identification algorithm in Section 2.Therefore, many sets of identified results can be obtained.Thestatistical parameters of the identified parameters can then beestimated by the statistical theory for example, the mean andstandard deviations of identified structural element stiffnesscan be calculated, respectively, by

𝜇

𝑖=

1

𝑛

𝑛

∑

𝑗=1

𝑘

𝑖𝑗; 𝜎

𝑖= √

1

𝑛 − 1

𝑛

∑

𝑗=1

(𝑘

𝑖𝑗− 𝑚

𝑖)

2

, (8)

in which 𝜇i and 𝜎i are the mean and standard deviations ofthe 𝑛-sets of identified stiffness of the 𝑖th structural element𝑘

𝑖, respectively.Then, a damage index𝐷

𝑖for the 𝑖th structural element is

defined as

𝐷

𝑖=

(𝜇

𝑑

𝑖− 𝜇

𝑢

𝑖)

𝜇

𝑢

𝑖

,

(9)

in which 𝜇𝑑𝑖and 𝜇𝑢

𝑖are the mean values of the identified 𝑖th

structural element stiffness in the damaged and undamagedstructure, respectively. Thus, the damage index 𝐷

𝑖tracks the

degrading of the identified 𝑖th structural element stiffness andcan also reflect its damage severity.

Analogously, the effect of uncertainties on the identifiedunknown excitation can be decreased by using the statisticalaverage of multisets of identified input time histories, that is,

f𝑢

=

1

𝑛

𝑛

∑

𝑗=1

f𝑢𝑗, (10)

where f𝑢

is the mean value of the 𝑛-sets that identifiedunknown excitation time histories and f𝑢

𝑗is the 𝑗th set of

identified unknown excitation.

3.2. The Identification Probability of Structural Damage.Structural damage is assumed as the degrading of the identi-fied 𝑖th structural element stiffness; a random variable of therelative change of the identified 𝑖th structural element stiff-ness in the damaged and undamaged structures is introducedas

𝑟

𝑖=

(𝑘

𝑑

𝑖− 𝑘

𝑢

𝑖)

𝑚

𝑢

𝑖

,

(11)

where 𝑘𝑑𝑖and 𝑘𝑢𝑖are the identified values of the 𝑖th structural

element in the damaged and undamaged structures, respec-tively.

Then, the probability of structural damages in this studyis estimated based on the reliability theory; that is, theprobability of structural damages of the 𝑖th structural element𝑃

𝐷𝑖is identified as

𝑃

𝐷𝑖= ∫

𝑟𝑖≤0

𝑝 (𝑟

𝑖) 𝑑𝑟

𝑖, (12)

where 𝑝(𝑟i) is the probability density function of the randomvariable 𝑟i in (11).

The random variable 𝑟i can be assumed as a normal ran-dom variable. Then, damage probability 𝑃

𝐷𝑖can be estimated

based on the definition of the standard normal distributionas

𝑃

𝐷𝑖= 1 − Φ(

𝜇

𝑟𝑖

𝜎

𝑟𝑖

) , (13)

where Φ(∙) denotes the probability of a standard normaldistribution.


Based on the probability 𝑃𝐷𝑖defined in (13), the 𝑃

𝐷𝑖value

presents the probability of whether the 𝑖th structural elementis damaged, and it is in the range of 50%–100%. A value of50% indicates that the structural element has no damage,whereas a value of larger than 50% means the occurrence ofdamage. The closer to 100% of the 𝑃

𝐷𝑖value, the larger the

damage probability.

4. A Numerical Simulation Example

In this paper, a numerical simulation example of the iden-tification of structural damage of a 10-story shear buildingmodel and its unknown excitation at the top floor is usedto demonstrate the efficiency of the proposed probabilisticapproach.The following structural parametric values are usedin the numerical study of the 10-story shear building: eachstory stiffness 𝑘

1= 𝑘

2= ⋅ ⋅ ⋅ = 𝑘

10= 6.79 × 10

3 kN/m,the concentrated mass at each floor level is 𝑚

1= 3.45 ×

10

3 kg, 𝑚2= 𝑚

3= 2.65 × 10

3 kg, 𝑚4= 𝑚

5= ⋅ ⋅ ⋅ = 𝑚

10=

1.81 × 10

3 kg. Rayleigh damping assumption is employed inthis study and the two Rayleigh damping coefficients are 𝛼 =2.88 and 𝛽 = 5.65.

The building is excited by a randomGaussian white noiseat the top floor; however, this excitation is assumed unknownin the identification process. Partial structural accelerationresponses at the 1st, 2nd, 3rd, 5th, 7th, 9th, and 10th floorlevels are used as the observation vector.

The uncertainties of measurement noises on the resultsof system identification are considered by superimpositionof noise process with the theoretically computed responsequantities, that is,

y𝑗= y𝑗0+ 𝜂

𝑗]𝑗𝜎 (y𝑗0) , (14)

where y𝑗and y

𝑗0are the 𝑗th set of measured acceleration

vector and calculated acceleration vector, respectively, ]𝑗

is the 𝑗th random vector with standard normal randomdistribution, 𝜎(y

𝑗0) is the standard deviation of the calculated

accelerations y𝑗0, and 𝜂

𝑗is the level of noise inmeasurements,

which is an important parameter representing the level ofuncertainties in the measured accelerations.

Structural damages of the building are assumed as follow:the 3rd story stiffness 𝑘

3is reduced by 5%, the 5th story

stiffness 𝑘5is reduced by 20%, and the 8th story stiffness 𝑘

8is

reduced by 10%.Each set of the measured acceleration responses with

uncertainties of measurement noise is used to identifystructural physical parameters, structural damage, and theunknown excitation to the building using the deterministicidentification algorithm in Section 2. In the probabilisticapproach as shown in Section 3, the measured accelerationscontaminated by noises are taken as the uncertain variablesas shown by (14). The Mote Carlo method is performed withsample size equal to 100. Two measurement noise levels,5% and 20%, are simulated to examine the effectiveness ofproposed algorithms.

Figures 1(a)-1(b) show the comparisons of the identifi-cation results by the deterministic identification algorithmand the probabilistic approach when the measurement noise

levels are equal to 5% and 20%, respectively. It is seenfrom Figure 1(a) that the identification values of the relativestiffness change by the deterministic and the probabilisticapproach are very close to the true values, indicating that bothapproaches can identify structural element damage when themeasurement noise level is low. However, Figure 1(b) showsthat when measurement noise level is quite high, which isequal to 20%, the identification error by the deterministicalgorithm increases and the false positives of damages occurin several undamaged floor stiffness, especially the falsedamage identification of about 7% reduction of k

1. On

the other hand, the damage index 𝐷𝑖in the probabilistic

approach can still accurately indicate the location and severityof structural damage as shown in Figure 1(b) and Table 1.Thisdemonstrates that the proposed probabilistic approach canavoid the false identification of damages by the deterministicalgorithm.

Figures 2(a)–2(c) compare the identification results ofunknown excitation by the deterministic algorithm and theprobabilistic approach. Form these comparisons, it is shownthat the identification accuracy on the unknown excitationby the deterministic algorithm decreases with the increase ofthe measurement noise level. There is an obvious deviationbetween the identified and the true excitations. However,Figure 2(c) demonstrates that the effect of uncertainties ofmeasurement noise on the identified unknown excitation canbe diminished by using the statistical average of multisets ofidentified unknown excitation time histories.

The identification results of structural damage probabili-ties defined in Section 3.2 for all the elements are summarizedin Table 1. It is shown that the damage probabilities ofelements 3, 5, and 8 are close to 100%, which are much largerthan 50%. The damage probabilities of all the undamagedelements are very close to 50%. This indicates that theproposed identification probability of structural damage canaccurately indicate all the structural damages locations.

5. Conclusions

In this paper, a probabilistic approach is proposed for theidentification of structural damage and unknown externalexcitations using only limited measurements of structuralacceleration responses contaminated by intensive measure-ment noises.The probabilistic approach is an improvement ofthe deterministic identification algorithm recently proposedby the authors. Structural parameters and unknown excita-tion are identified in a sequential manner, which simplifiesthe identification problem compared with other simulta-neous identification algorithms. The statistical parametersof the identified structural parameters are estimated usingthe statistical theory, and a damage index is defined toindicate the location and severity of structural damage.The probability of identified structural damage is furtherderived based on the reliability theory.The unknown externalexcitation on the structure can also be derived by statisticalaverage of multisets of identified unknown excitation time-histories. Therefore, the novelty of the research is that it pro-poses a probabilistic approach which can accurately identify


1 2 3 4 5 6 7 8 9 10Element number

Rela

tive s

tiffne

ss ch

ange

(%)

With 5% noise

TrueDeterministic methodProbabilistic method

−20

−15

−10

−5

0

5

(a) Comparisons of identified stiffness with 5% measurement noise

1 2 3 4 5 6 7 8 9 10Element number

With 20% noise

TrueDeterministic methodProbabilistic method

Rela

tive s

tiffne

ss ch

ange

(%)

−20

−15

−10

−5

0

5

(b) Comparisons of identified stiffness with 20% measurement noise

Figure 1: Comparisons of identification stiffness with different measurement noise levels.

0 0.5 1 1.5 2 2.5 3 3.5 4

0

200

400

600

Time (s)

Forc

e (N

)

With 5% noise

TrueDeterministic method

−400

−200

(a) Identified unknown excitation by the deterministic algorithm with 5%noise

0 0.5 1 1.5 2 2.5 3 3.5 4

0

500

1000

1500

Time (s)

Forc

e (N

)

With 20% noise

−500

TrueDeterministic method

(b) Identified unknown excitation by the deterministic algorithm with20% noise

0 0.5 1 1.5 2 2.5 3 3.5 4

0

500

Time (s)

Forc

e (N

)

With 20% noise

−500

TrueProbabilistic method

(c) Identified unknown excitation by the probabilistic approach with 20%noise

Figure 2: Comparison of identified unknown excitation.


Table 1: Comparisons of identified damage indices and damage probabilities.

Storyno. Actual damage (%)

Noise level5% 20%

𝐷

𝑖(%) Error (%) 𝑃

𝑖(%) 𝐷

𝑖(%) Error (%) 𝑃

𝑖(%)

1 0.00 0.01 0.01 49.2 0.34 0.34 49.42 0.00 −0.58 0.58 50.3 −0.97 0.97 56.63 −5.00 −4.32 0.68 99.2 −4.27 0.73 90.84 0.00 −0.45 0.45 50.3 −0.51 0.51 54.25 −20.00 −19.32 0.68 100 −18.63 1.37 1006 0.00 0.01 0.01 49.9 −0.53 0.53 55.77 0.00 −0.900 0.900 51.8 −1.30 1.30 61.48 −10.00 −9.00 1.00 100 −8.58 1.41 92.19 0.00 −0.58 0.58 50.7 −0.67 0.67 56.510 0.00 −0.75 0.75 51.4 −0.73 0.73 57.2Total error — — 0.88 — 0.85

structural damage and the unknown excitations more thanthe deterministic identification algorithm under high-levelmeasurement noises. The proposed probabilistic approachis clear and simple compared with other previous algo-rithms. A numerical simulation example demonstrates thatthe proposed probabilistic approach can accurately identifystructural damage and the unknown excitations using onlypartial measurements of structural acceleration responsescontaminated by intensive measurement noises.

It is important to investigate the efficiency of the proposedprobabilistic approach for the identification of other typesof structural systems. Moreover, damage identification isonly verified by the numerical simulation in this paper.Experimental studies to fully assess the performances of theproposed algorithm are needed. Such work is investigated bythe authors and the results will be reported in future.

Acknowledgments

This research is funded by the National Natural ScienceFoundation of China (NSFC) through Grant no. 51178406,the research funding SLDRCE10-MB-01 from the State KeyLaboratory for Disaster Reduction in Civil Engineering atTongji University, China, and by the Fujian Natural ScienceFoundation through Grant no. 2010J01309.

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A Probabilistic Damage Identification Approach for ...In this paper, a probabilistic approach is proposed for the identification of structural damage and unknown external excitations