A Probabilistic Damage Identification Approach for ...In this paper, a probabilistic approach is...
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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 ยฉ 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 [1โ5]. 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 [22โ26]. 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 (๐ก) + Cx (๐ก) + Kx (๐ก) = B๐ขf๐ข(๐ก) , (1)
in which x(๐ก), x(๐ก), and x(๐ก) are the vectors of displacement,velocity, and acceleration response, respectively; M, C, andK are the mass, damping, and stiffness matrices, respectively;f๐ข(๐ก) is an unmeasured external excitation vector; and B๐ข isthe influence matrix associated with f๐ข(๐ก). 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 = [X๐1,X๐2, ๐๐
]
๐
; X1= x;
X2= x; ๐
๐
= [๐
1, ๐
2, . . . , ๐
๐]
๐
,
(2)
where ๐๐ is a vector of the ๐-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, f๐ข) . (3)
Usually, only a limited number of accelerometers aredeployed in structures to measure acceleration responses.Therefore, the discretized observation equation can beexpressed as
y [๐] = h (X [๐]) + G๐ขf๐ข [๐] + v [๐] , (4)
where G๐ข = DMโ1B๐ข; h(X[๐]) = DMโ1{โ(C๐)X2[๐] โ (K)๐X1[๐]} in which y[๐] is observation vector (measured accel-
eration responses) at time ๐ก = ๐ ร ฮ๐ก with ฮ๐ก being thesampling time step, (C)๐ represents elements in the dampingmatrix C composed by the unknown parameters of dampingin the parametric vector ๐, (K)๐ represents the constitution ofstiffness matrixK analogously, f๐ข[๐],X
1[๐] and X
2[๐] are the
corresponding discretized values at time ๐ก = ๐ ร ฮ๐ก, D is thematrix associated with the locations of accelerometers, andk[๐] is the measured noise vector assumed to be a Gaussianwhite noise vector with zero mean and a covariance matrixE[v๐v๐๐] = R๐๐๐ฟ
๐๐, where ๐ฟij is the Kronecker delta.
Based on the extended Kalman estimator [15, 16], theextended state vector at time ๐ก = (๐+1)รฮ๐ก can be estimatedwith the observation of (y[1], y[2], . . . , y[๐]) as follows:
X [๐ + 1 | ๐] = X [๐ + 1 | ๐] + K [๐]
ร {y [๐] โ โ (X [๐ | ๐ โ 1] , f [๐])
โG๐ขf๐ข [๐ | ๐]} ,
(5)
in which
X [๐ + 1 | ๐] = X [๐ | ๐ โ 1] + โซ๐ก[๐+1]
๐ก[๐]
g (X, f๐ข) ๐๐ก, (6)
where X[๐+1 | ๐] and f๐ข[๐ | ๐] are the estimation ofX[๐+1]and f๐ข[๐] given (y[1], y[2], . . . , y[๐]), respectively, andK[๐]is the Kalman gain matrix [15, 16].
However, since the external excitation f๐ข(๐ก) 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 f๐ข(๐ก) acts; that is, matrix G๐ข in (4) is nonzero; theunknown excitations at time ๐ก = (๐+1)รฮ๐ก can be estimatedfrom (4) by the least-square estimation as [15, 16]
f๐ข [๐ + 1 | ๐ + 1] = [(G๐ข)๐G๐ข]โ1
(G๐ข)๐
ร {y [๐ + 1] โ h (X [๐ + 1 | ๐])} ,(7)
Journal of Applied Mathematics 3
in which f๐ข[๐ + 1 | ๐ + 1] is the estimation of f๐ข[๐ + 1] giventhe observation of (y[1], y[2], . . . , y[๐ + 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
๐
๐=
1
๐
๐
โ
๐=1
๐
๐๐; ๐
๐= โ
1
๐ โ 1
๐
โ
๐=1
(๐
๐๐โ ๐
๐)
2
, (8)
in which ๐i and ๐i are the mean and standard deviations ofthe ๐-sets of identified stiffness of the ๐th structural element๐
๐, respectively.Then, a damage index๐ท
๐for the ๐th structural element is
defined as
๐ท
๐=
(๐
๐
๐โ ๐
๐ข
๐)
๐
๐ข
๐
,
(9)
in which ๐๐๐and ๐๐ข
๐are the mean values of the identified ๐th
structural element stiffness in the damaged and undamagedstructure, respectively. Thus, the damage index ๐ท
๐tracks the
degrading of the identified ๐th 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,
f๐ข
=
1
๐
๐
โ
๐=1
f๐ข๐, (10)
where f๐ข
is the mean value of the ๐-sets that identifiedunknown excitation time histories and f๐ข
๐is the ๐th set of
identified unknown excitation.
3.2. The Identification Probability of Structural Damage.Structural damage is assumed as the degrading of the identi-fied ๐th structural element stiffness; a random variable of therelative change of the identified ๐th structural element stiff-ness in the damaged and undamaged structures is introducedas
๐
๐=
(๐
๐
๐โ ๐
๐ข
๐)
๐
๐ข
๐
,
(11)
where ๐๐๐and ๐๐ข๐are the identified values of the ๐th 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 ๐th structural element๐
๐ท๐is identified as
๐
๐ท๐= โซ
๐๐โค0
๐ (๐
๐) ๐๐
๐, (12)
where ๐(๐i) is the probability density function of the randomvariable ๐i in (11).
The random variable ๐i can be assumed as a normal ran-dom variable. Then, damage probability ๐
๐ท๐can be estimated
based on the definition of the standard normal distributionas
๐
๐ท๐= 1 โ ฮฆ(
๐
๐๐
๐
๐๐
) , (13)
where ฮฆ(โ) denotes the probability of a standard normaldistribution.
4 Journal of Applied Mathematics
Based on the probability ๐๐ท๐defined in (13), the ๐
๐ท๐value
presents the probability of whether the ๐th structural elementis damaged, and it is in the range of 50%โ100%. 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 ๐
๐ท๐value, 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 ๐
1= ๐
2= โ โ โ = ๐
10= 6.79 ร 10
3 kN/m,the concentrated mass at each floor level is ๐
1= 3.45 ร
10
3 kg, ๐2= ๐
3= 2.65 ร 10
3 kg, ๐4= ๐
5= โ โ โ = ๐
10=
1.81 ร 10
3 kg. Rayleigh damping assumption is employed inthis study and the two Rayleigh damping coefficients are ๐ผ =2.88 and ๐ฝ = 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,
y๐= y๐0+ ๐
๐]๐๐ (y๐0) , (14)
where y๐and y
๐0are the ๐th set of measured acceleration
vector and calculated acceleration vector, respectively, ]๐
is the ๐th random vector with standard normal randomdistribution, ๐(y
๐0) is the standard deviation of the calculated
accelerations y๐0, and ๐
๐is 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 ๐
3is reduced by 5%, the 5th story
stiffness ๐5is reduced by 20%, and the 8th story stiffness ๐
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 ๐ท๐in 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)โ2(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
Journal of Applied Mathematics 5
1 2 3 4 5 6 7 8 9 10Element number
Rela
tive s
tiffne
ss ch
ange
(%)
With 5% noise
TrueDeterministic methodProbabilistic method
โ20
โ15
โ10
โ5
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
(%)
โ20
โ15
โ10
โ5
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
โ400
โ200
(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
โ500
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
โ500
TrueProbabilistic method
(c) Identified unknown excitation by the probabilistic approach with 20%noise
Figure 2: Comparison of identified unknown excitation.
6 Journal of Applied Mathematics
Table 1: Comparisons of identified damage indices and damage probabilities.
Storyno. Actual damage (%)
Noise level5% 20%
๐ท
๐(%) Error (%) ๐
๐(%) ๐ท
๐(%) Error (%) ๐
๐(%)
1 0.00 0.01 0.01 49.2 0.34 0.34 49.42 0.00 โ0.58 0.58 50.3 โ0.97 0.97 56.63 โ5.00 โ4.32 0.68 99.2 โ4.27 0.73 90.84 0.00 โ0.45 0.45 50.3 โ0.51 0.51 54.25 โ20.00 โ19.32 0.68 100 โ18.63 1.37 1006 0.00 0.01 0.01 49.9 โ0.53 0.53 55.77 0.00 โ0.900 0.900 51.8 โ1.30 1.30 61.48 โ10.00 โ9.00 1.00 100 โ8.58 1.41 92.19 0.00 โ0.58 0.58 50.7 โ0.67 0.67 56.510 0.00 โ0.75 0.75 51.4 โ0.73 0.73 57.2Total error โ โ 0.88 โ 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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