Assessment of damage indicators for structural health ...paginas.fe.up.pt › ~eurodyn2014 › CD › papers › 318_MS13... · Such vibration-based SHM systems consist of a data

Assessment of damage indicators for structural health monitoring using aprobabilistic frameworkMaik Brehm1, Arnaud Deraemaeker1

1Universite libre de Bruxelles, BATir - Structural and Material Computational Mechanics,50 av F.D. Roosevelt, CP 194/2, 1050 Bruxelles, Belgium

email: [email protected], [email protected]

ABSTRACT: Civil and mechanical structures are exposed to natural ageing and accidental overloading. To increase safety andreduce costs simultaneously, current classical maintenance strategies can be supported by advanced fully automatic vibration-basedstructural health monitoring (SHM) systems.

Such vibration-based SHM systems consist of a data acquisition system with sensors applied on a structure. From recordedtime histories, damage indicators are generated and can be monitored in control charts. Sensitivity to damage and insensitivity toconfounding effects are essential properties for a well-designed indicator. To design such damage indicators, experimental tests areinevitable. As such tests can be expensive with respect to time and labor costs, researchers are interested in replacing these testsby numerical simulations. Recently, a semi-analytical probabilistic framework has been proposed by the authors to determine thestatistical properties of damage indicators based on the known statistical properties of a random excitation in the time domain anda finite element model. Therefore, a time consuming numerical time integration scheme, as typically applied for each sample of asample-based stochastic structural analysis, can be avoided.

In the presented paper this probabilistic framework is applied to compare efficiently two different damage indicators basedon modal filters using a numerical cracked concrete beam model. For this numerical example, the statistical properties of thedamage indicators will be investigated with progressing damage. From these statistical properties, it can be determined underwhich conditions the distribution of the damage indicators can be assumed to be normal and, therefore, if the application of certaincontrol charts is justified.

KEY WORDS: Uncertainty quantification, damage detection, structural health monitoring, structural dynamics, random vibrations

1 INTRODUCTION

Vibration-based structural health monitoring can be a veryeffective tool to support efficiently traditional maintenancestrategies, and therefore reduce operational costs of civil ormechanical structures and systems. An advantage of vibration-based structural health monitoring (SHM) in comparison toother structural health monitoring systems, is the use of ambientexcitation sources to introduce energy into the structure. Hence,no additional artificial excitation source is needed, which allowskeeping the structure in operation during investigations.

Such a vibration-based SHM system consists of a dataacquisition system with sensors applied on a structure. Therecorded time histories are postprocessed and features areextracted to construct damage indicators. These damageindicators are then monitored over the life time of the structureusing for example control charts, from which alerts are triggeredif certain control limits are crossed. Finally, the alerts initiate thedecision making. Therefore, the damage indicators are a veryimportant part of the whole structural monitoring system. Theyneed to be sensitive to damage and insensitive to confoundingeffects, like measurement noise, operational, and environmentalconditions. In addition, several control charts (e.g., X-bar,Hotelling-T2) require normally distributed damage indicatorsto set meaningful control limits. Due to the variation ofthe excitation for each time interval and random measurement

errors, responses and respectively damage indicators are timevariant even if damage is not propagating. Consequently, thestatistics of damage indicators as a function of the statisticsof random excitations and measurement errors are importantfor the definition of control limits and the assessment of thesuitability of damage indicators.

The motivation of this paper is the investigation of thestatistical properties of two indicators based on modal filters,Ip and m4, proposed in [1][2][3] and applied on laboratorystructures in [4][5], with respect to the time length of measuredresponse time histories and intensity of measurement errors.Traditionally, such investigations were performed by the useof experimental data gained from expensive testing campaigns[6]. More and more, such experimental tests have been replacedby numerical simulation strategies using, for example, intrinsicMonte Carlo sampling coupled with a time integration scheme[7] or more advanced with the application of meta models [8].

In this paper, a recently proposed approach for uncertaintyquantification [9][10] is applied to derive analytically thestatistics of linearly combined response Fourier transforms.Based on these statistics, the statistics of damage indicators areobtained by generating and evaluating Latin hypercube samples.The numerical representation of the structure is assumed to bedeterministic. To create realistic damage patterns, the resultsof the damage propagation calculations performed in [11] are

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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reused. Therein, the constitutive law parameters have beendetermined by an optimization-based model calibration strategy[12].

For the variation of time length and measurement errorintensity, the investigations of the probability density functionsof the two damage indicators show a very different behavior. Byassessing the performance of the indicator with the false positiveprobability, strong similarities between both indicators can beobserved. Finally, both indicators are equally capable to identifysmall damages.

2 UNCERTAINTY QUANTIFICATION OF DISCRETE RE-SPONSE FOURIER TRANSFORMS

This section reviews the main theoretical aspects of theprobabilistic framework introduced in [10], applied in this paper.

2.1 General concept and problem description

Assuming a set of continuous response time signals x(t) ∈ Rmx

for mx degrees of freedom over time t of a structure is given by

x(t) = xf(t)+xn(t). (1)

The random continuous time signal xf(t) ∈ Rmx is the trueerrorless response resulting from a random continuous weaklystationary non-periodic excitation signal f(t) ∈ Rm f at m fdegrees of freedom. The random continuous time signal xn(t) ∈Rmx is the weakly stationary measurement error at the respectivedegrees of freedom. In practical applications, many pre- andpostprocessing techniques are applied on the measured responsesignal. A typical operator is a linear combiner, which is definedas

g(t) = Ax(t) = A xf(t)︸︷︷︸gf(t)

+A xn(t)︸︷︷︸gn(t)

(2)

in the time domain, with g(t) ∈ Rmg and the time-invariantmatrix of linear combination coefficients A ∈ Rmg×mx . Inaddition, windowing in the time domain is often applied priorto a further signal processing, such as Fourier transformation.

From now on, the random continuous excitation f(t) isassumed to be a multivariate normal distribution with knownexpectations and covariances. By extracting a finite discreterandom excitation signal f j from f(t) with N discrete valuesfor each excited degree of freedom j = 1,2, . . . ,m f , thecorresponding random discrete vector

f =[

fT1 fT

2 · · · fTm f

]T(3)

of dimension Nm f can be defined as multivariate normallydistributed

f∼N(E(f),C(f, f)

)(4)

with an expectation vector E(f)∈RNm f and a covariance matrixC(f, f) ∈ RNm f×Nm f , including correlations in time and space.

The random continuous signal of measurement errors xn(t)is also assumed to be multivariate normally distributed. Thefinite discrete random errors xn j obtained from xn(t) with Ndiscrete values for each measured response degree of freedomj = 1,2, . . . ,mx are collected in a random discrete vector

xn =[

xnT1 xn

T2 · · · xn

Tmx

]T. (5)

This random vector can be described as normally distributed

xn ∼N (E(xn),C(xn, xn)) (6)

with the expectation vector E(xn) ∈ RNmx and the covariancematrix C(xn, xn) ∈ RNmx×Nmx , which takes correlations in timeand space into account. In this paper, it is assumed thatmeasurement errors and excitations are independent from eachother.

To derive the statistics of the discrete Fourier transform of alinear combiner of responses based on the known statistics of theexcitations and the measurement errors, a linear deterministicoperator Zf is derived to perform analytically the uncertaintypropagation between the random excitations in the time domainand the response Fourier transforms of the linear combiner. Theuncertainty propagation of the measurement errors defined inthe time domain can also be realized by a linear operator Zn.

The mean value of the discrete Fourier transform of theerrorless linear combiner can be obtained by

E([

Re(F gf

)Im(F gf

) ])=

[Re(Zf)

Im(Zf) ] E

(f)

(7)

and its corresponding covariance matrix by

C([

Re(F gf

)Im(F gf

) ] ,[ Re(F gf

)Im(F gf

) ])=

[Re(Zf)

Im(Zf) ]C

(f, f)[ Re

(Zf)

Im(Zf) ]T

,

(8)

where Re(·) and Im(·) are the real and imaginary partof a complex scalar, vector, or matrix. In analogy,

the mean value E([

Re(F gn)Im(F gn)

])and covariance matrix

C([

Re(F gn)Im(F gn)

],

[Re(F gn)Im(F gn)

])of the measurement errors

can be obtained applying x and Zn instead of f and Zf. Asthe measurement errors are assumed to be independent from theerrorless response signal and the excitations, the statistics of thediscrete Fourier transform of the linear combiner of the noise-disturbed discretized signal g is obtained by

E([

Re(F g)Im(F g)

])=E

([Re(F gf

)Im(F gf

) ])+E([

Re(F gn)Im(F gn)

])(9)

and

C([

Re(F g)Im(F g)

],

[Re(F g)Im(F g)

])= C

([Re(F gf

)Im(F gf

) ] ,[ Re(F gf

)Im(F gf

) ])+C

([Re(F gn)Im(F gn)

],

[Re(F gn)Im(F gn)

]).

(10)

A detailed description of the assembly of the vector g and itsdiscrete Fourier transform F g will be given in the followingsubsections. In this paper, discretization errors are assumed tobe negligible by choosing a sufficiently small time step.

In the following subsections, the derivations of the linearoperators, Zn and Zf, are explained in detail. The time-invariantmodal system properties are assumed to be deterministic.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

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2.2 Analytical uncertainty propagation for errorless signal

Using Duhamel’s integral for multiple inputs and multipleoutputs given in [13, p.95], the linear combination of thecontinuous response displacements in the time domain of aproportional viscously damped linear system under continuousexcitation f(t) ∈ Rm f for t ≥ 0 can be obtained by

gf(t) =t∫

0

h(t− τ)f(τ)dτ (11)

with gf(t) ∈ Rmg and the matrix of linear combinationcoefficients A ∈ Rmg×mx . The impulse response function h(t) ∈Rmg×m f is given by

h(t) = AΦΦΦxd(t)ΦΦΦfT, (12)

where d(t) ∈ Rmλ×mλ represents the time dependent diagonalmatrix with diagonal elements

(d(t))l,l =sin(√(λλλ )l(1− (ζζζ )2

l )t)√(λλλ )l(1− (ζζζ )2

l )exp(−

√(λλλ )

l(ζζζ )lt). (13)

It is assumed that the modal properties of the system are timeinvariant. The modal properties for mλ considered modes arethe classical undamped eigenvalues λλλ ∈Rmλ , the correspondingmodal damping ratios ζζζ ∈ Rmλ , and the eigenvector matrix ΦΦΦ.The mode shape matrices of response degrees of freedom ΦΦΦx ∈Rmx×mλ and of excitation degrees of freedom ΦΦΦf ∈ Rm f×mλ areassembled from the mass normalized eigenvector matrix ΦΦΦ.

From the continuous infinite linear combiner g(t), a finitetime frame is extracted through a window function w(t) withfinite compact support within ts < t ≤ te. The application of thiswindow function leads to the response of the ith linear combinerof the finite time frame

gwi(t) = w(t)m f

∑j=1

t∫0

(h(t− τ))i, j (f(τ)) jdτ. (14)

By introducing a time step ∆t and defining ts = (s− 1) ∆t andte = e ∆t with the positive integer values s,e ∈ Z and 0 < s < e,the discrete form of Eq. (14)

gwi = w◦m f

∑j=1

qi, j f j (15)

is derived in the time step interval [s,e] with gwi ∈Rmw and mw =e− s+ 1. The symbol ◦ denotes the Schur product (e.g., [14]).The vector w ∈Rmw represents the discretization of the windowfunction w within the support [s,e]. The vector of the jth degreeof freedom of the excitation f j ∈RN is given for all discrete timesteps [1,N] with N = e. The matrix qi, j ∈Rmw×N is derived fromthe impulse response function related to the linear combiner i

and the excitation degree of freedom j.

qi, j = ∆t

hi, j,s hi, j,s−1 hi, j,s−2 . . . 0hi, j,s+1 hi, j,s hi, j,s−1 . . . 0hi, j,s+2 hi, j,s+1 hi, j,s . . . 0... hi, j,s+2 hi, j,s+1 . . . 0

hi, j,e−2... hi, j,s+2 . . . 0

hi, j,e−1 hi, j,e−2...

. . . 0hi, j,e hi, j,e−1 hi, j,e−2 . . . hi, j,e−N+1

∀ n≤ 0 : hi, j,n = 0

(16)In a next step the discrete Fourier transformation is applied to

Eq. (15) through a complex matrix operator B ∈ C(mw2 +1)×mw

containing the coefficients [15, p. 50]

(B)k,n = ∆t exp(−ι

2π

N(k−1)(n−1)

)(17)

with the imaginary unit ι =√−1 for n = 1,2, . . . ,mw and k =

1,2, . . . , mw2 +1. This yields

F gw i = B

(w◦

m f

∑j=1

qi, j f j

)(18)

with F gw i ∈ C(mw2 +1). Eq. (18), which represents the Fourier

transform of the ith linear combination for windowed responses,can be reformulated to

F gw i =

m f

∑j=1

Zfi, j f j (19)

usingZfi, j = B

((11×N ⊗w

)◦qi, j

), (20)

where ⊗ denotes the Kronecker product. The matrix 11×N is aninteger matrix of dimension 1×N only filled with the value 1.

By combining the evaluations of Eq. (19) for all linearcombinations i = 1,2, . . . ,mg and all excitation degrees offreedom j = 1,2, . . . ,m f in the random vector and lineardeterministic operator,

F gf =

F gw 1F gw 2

...F gw mg

(21)

and

Zf =

Zf1,1 Zf1,2 . . . Zf1,m f

Zf2,1 Zf2,2 . . . Zf2,m f...

.... . .

...Zfmg,1 Zfmg,2 . . . Zfmg,m f

, (22)

respectively, a simple linear relation

F gf = Zf f with Zf ∈ C(mw2 +1)mg×Nm f (23)

between the random discrete excitation time series f definedin Eq. (4) and the discrete Fourier transform of the linearlycombined windowed response displacement signals representedby F gf can be derived.


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2.3 Analytical uncertainty propagation for measurementerrors

The ith linear combiner related to the measurement errors isgiven by

gnwi =mx

∑j=1

(A)i, jqw xn j (24)

with gnwi ∈ Rmw . The vector xn j ∈ RN represents themeasurement error of the jth degree of freedom of the responsesignal. The matrix qw ∈ Rmw×N is defined as

qw =[

0mw×s−1 wImw], (25)

where Imw represents the identity matrix of dimension mw. Thevector w is assembled by the discrete values of the supportvalues of the window function w(t) as defined in Subsection2.2. The matrix 0mw×s−1 is a zero valued matrix of dimensionmw× s− 1. The expression qwxn j is the windowed time frameof the measurement errors at the jth degree of freedom.

Similar to Section 2.2, the matrix of discrete Fouriercoefficients B is applied to Equation (24) to obtain thediscrete Fourier transform of linearly combined windowedmeasurements errors

F gnw i = Bmx

∑j=1

(A)i, j qw xn j, (26)

which can be rewritten as

F gnw i =mx

∑j=1

Zni, j xn j (27)

withZni, j = (A)i, j B qw, (28)

where F gnw i ∈ C(mw2 +1) and Zni, j ∈ C(mw

2 +1)×N .The expressions

F gn =

F gnw 1F gnw 2

...F gnw mg

(29)

and

Zn =

Zn1,1 Zn1,2 . . . Zn1,mxZn2,1 Zn2,2 . . . Zn2,mx

......

. . ....

Znmg,1 Znmg,2 . . . Znmg,mx

(30)

can be derived by evaluating Equation (27) for all possible linearcombinations i = 1,2, . . . ,mg and all possible response degreesof freedom j = 1,2, . . . ,mx.

Finally, a linear combination

F gn = Zn xn with Zn ∈ C(mw2 +1)mg×Nmx (31)

can be found that relates the random measurement errors of theresponses indicated by xn ∈RNmx defined in Equation (5) to thediscrete Fourier transform of the linearly combined windowedmeasurement errors summarized in F gn ∈ C(mw

2 +1)mg .

3 DAMAGE DETECTION WITH PEAK INDICATORS

3.1 General concept

As shown in [7], the application of modal filters combined withpeak indicators is very efficient to detect early damage usingmeasured vibration data. The modal filters are designed withrespect to a certain mode by defining the coefficients of a linearcombiner matrix A in order to eliminate the remaining modeswithin a certain frequency range of the Fourier transforms.Hence, only one significant peak is visible in the power spectraldensities of the undamaged structure. The linear combinationcoefficients are defined once for the undamaged or referencestate. With increasing damage, peaks at the eigenfrequenciesof the modes previously eliminated appear. This appearancecan be detected for each time frame of a vibration signal by apeak indicator. Figure 1 shows qualitatively the power spectraldensity of linearly combined responses. The applied modalfilter was designed for the second mode of a structure. Dueto the presence of random measurement errors and randomexcitation, the peak indicators are not constant over time (i.e.not deterministic) and can be characterized by its empiricalprobability density function. If a damage occurs, a changeof the statistical properties can be observed, which can bemonitored in control charts. The exceedance of control limitsindicates a possible damage and initiates further decisionmaking processes.

In order to obtain the statistics of the peak indicator for acertain load step within a virtual testing scheme, as foreseenin this paper, the following procedure is applied. First, theapproach explained in Section 2 is used to define analyticallythe statistics of the Fourier transforms of linearly combinedresponses. From this multivariate distribution, Latin hypercubesample sets are generated to compute sample sets of powerspectral densities. For each sample set of power spectraldensities, one sample of a peak indicator can be obtained.Using all samples of the peak indicator an estimation of theprobability density function and the corresponding statistics canbe obtained. The aim of this paper is to investigate the suitabilityof two different peak indicators, Ip and m4, to detect smalldamages.

10−11

10−10

10−9

10−8

10−7

√(λλλ )1

√(λλλ )2

√(λλλ )3

ka kb kc kd

pow

ersp

ectr

alde

nsity

S

circular frequency ω

discrete frequency number k

undamageddamaged

Figure 1: Example of a power spectral density of a linearcombiner using a modal filter tuned to the second eigenmodeof the system. The grey frames indicate the frequency rangesused to calculate the peak indicators.


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3.2 Peak indicator Ip

In [16] a peak indicator was proposed that interprets anyfunction of interest S with discrete equidistant values (S )kdefined in the interval [ka,kb] as a discrete probability densityfunction. In this application S corresponds to the powerspectral density (see Figure 1). Refs. [2][3] applied successfullythis indicator in the context of damage detection.

According to [16], the peak indicator

Ip = σ2k

12(kb− ka)2 (32)

is derived from the quotient between the variance σ2k of the

investigated probability density function S within the interval[ka,kb] and the variance of a fitted uniform distribution, which

is equal to (kb−ka)2

12 . Consequently, if the function S is constantwithin [ka,kb] the indicator will be equal to one. Otherwise,the indicator is lower than one. Investigations of [16] showedthat the indicator is close to one, even if the function S is notconstant in the interval [ka,kb], but follows a monotonic trend.Higher values than one are possible, if a peak appears near theboundaries ka or kb of the interval.

The mean value of a discrete distribution calculated from anonnormalized discrete probability density function S definedin the interval [ka,kb] is given by

µk =kb

∑k=ka

k (S )k

(kb

∑k=ka

(S )k

)−1

. (33)

The second centered statistical moment, which is the variance,yields

σ2k =

kb

∑k=ka

(k−µk)2(S )k

(kb

∑k=ka

(S )k

)−1

. (34)

3.3 Peak indicator m4

An alternative to the indicator Ip is the indicator m4, whichhas been originally introduced in [17] and has been adaptedby [3][4] for the detection of damage using modal filters. Thelower limit of this peak indicator is zero. The indicator tends toincrease with increasing damage.

In contrast to the indicator Ip, the discrete values (S )k ofa discrete function are interpreted directly as a set of samples.Their first order central moment

µs =1

kb− ka +1

kb

∑k=ka

(S )k (35)

and respective fourth order central moment

κs =1

kb− ka +1

kb

∑k=ka

((S )k−µs)4 (36)

are given. A scaling of κs has been introduced by [2] to decreasethe sensitivity with respect to different loading intensities.Finally, the normalized indicator m4 can be written as

m4 = (κs)14

(kd

∑k=kc

(S )k ∆ω

)−1

(37)

with ∆ω being the discrete frequency step.

4 BENCHMARK STUDY: SIMPLY SUPPORTED BEAM

4.1 Static loading and damage progress

In [11] a simply supported plain concrete beam has been studiedunder quasi static loading with progressing damage using animplicit gradient damage law [18]. The material and damagelaw parameters have been obtained from a model calibration[12]. Hence, the results of this study with respect to the damagepatterns are very realistic, which is the motivation to use themin the subsequent investigation of damage indicators.

The geometry of the beam and the loading configuration ispresented in Figure 2. The red patterns in Figure 2 describe apredamage (50% of original tensile strength) to enable multiplecracking of the beam. The load-deflection-curve for a three-point-bending test with measured displacements in verticaldirection at the position of the loading is shown in Figure 3with indications of the load steps (LS). A first crack growth hasbeen observed at LS 17. By means of two selected load steps,the evolution of the damage pattern with progressing damage isillustrated in Figure 4. The damage severity related to LS 50 canbe considered as small.

For the numerical simulations, the structure is modeled in 2Dwith a plane stress formulation and 9-node rectangular elementsof side length between 2.5 and 5mm.

P

50

100

1000[mm] 2000

x 1000 50

Figure 2: Geometry of investigated simply supported plainconcrete beam with static loading.

0

200

400

600

800

1000

0 0.2 0.4 0.6 0.8 1 1.2

load

P[N

]

deflection [mm]

LS 15

LS 25

LS 35

LS 45

LS 55LS 65

LS 115LS 17 (first crack appearance)

Figure 3: Load-deflection-curve with indication of load steps(LS).

LS 50

LS 60

2.00.0 1.0[1010]

Figure 4: Young’s modulus distribution around the cracks usingthe implicit damage law for load steps (LS) 50 and 60.


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4.2 Dynamic loading and response

The first three circular bending frequencies of the undamagedstructure are 227.1 rad

s , 874.8 rads , and 1703.7 rad

s . The modaldamping coefficient are assumed to be damage invariant andare set to 2.5% for each mode, which is a typical value forconcrete. For the dynamic calculation, the damaged systemsat each load step are linearized by extracting the global secantstiffness matrix.

The linear systems are excited by a random excitationintroduced at five positions in vertical direction as shownin Figure 5. The excitation is realized by independent andidentically distributed (i.i.d.) random variables with respectto time and space following a normal distribution with meanvalue zero and variance 218N2. The unit of the (co)variances ofexcitation is Newton to the power of two

[N2]. Therefore, the

random excitation can be described by

f∼N(E(f),C(f, f)

)(38)

with(E(f)

)i = 0 and

(C(f, f)

)i, j = δi, j218N2 ∀i, j (39)

using the Kronecker delta

δi, j =

{0 : i 6= j1 : i = j . (40)

The resulting strains of the structure are measured by anembedded chain of three fiber bragg grating sensors (FBGS)with a measurement length of 5cm each. The location is givenin Figure 5. It is assumed that the level of dynamic excitation islow enough to avoid a further damage of the structure, but highenough to obtain a sufficient resolution of the measured strains.The excitation leads after 2s to response strain distributionswith constant properties of a zero mean value and a varianceof approximately 10−8 for the undamaged system. For load step65 a strain variance of about 1.25 ·10−8 has been observed.

The measurement errors are defined as i.i.d. random variablesconnected directly to the three measured strains of the responseswith respect to time and space. They follow a normaldistribution with a zero mean value and a variance of ν22.5 ·10−11. The intensity factor of measurement errors ν is variedin Subsection 4.4 and is set to one if not otherwise stated.Consequently, the measurement errors can be described by

xn ∼N (E(xn),C(xn, xn)) (41)

with

(E(xn))i = 0 and (C(xn, xn))i, j = δi, jν22.5 ·10−11 ∀i, j.

(42)

1000 100050 502000

x

f2(t) f3(t)

200 200 200 200

f1(t) f4(t) f5(t)

955

200

[mm]

FBGS chain (3x5cm)

Figure 5: Geometry of investigated simply supported plainconcrete beam with dynamic loading.

The statistics of the measurement errors are identical for alldamage levels. For ν = 1 a signal-to-noise ratio between 400(undamaged) and 500 (LS 65) can be expected. In other words,the standard deviation of the measurement error is about 5%resp. 4.5% of the standard deviation of the errorless responsesignal, which are realistic values.

From the generalized eigenvalue problem the mode shapevectors related to the displacements can be extracted for eachlinearized system. The modal strains can be easily obtainedby multiplying these mode shape vectors by a transformationmatrix. By tuning the modal filter to the first and second bendingmode shape, the matrix of linear combination coefficients A isfinally given by

A =

[−1.4016 2.7383 −1.3868−0.2012 0.3291 −0.1289

]103, (43)

where the first and second row correspond to the coefficients ofthe first linear combiner (lc1) and second linear combiner (lc2),respectively. To calculate the statistics of the linear combiner ofresponse Fourier transforms according to Section 2, the first 10modes (mλ = 10) until a circular frequency of about 10000 rad

sand a time step ∆t = 2−11s are chosen. All investigationsconsider a time frame that is extracted by a rectangular windowfunction after the steady state has been reached (i.e. after astart time ts = 2s). In the focus of subsequent studies is theposition around the first eigenfrequency of the second linearcombiner. As the investigated peak indicators are designed todetect the early damage and therefore early peak appearance,only damages until load step 65 are considered.

In the following, the effects of time length and measurementerror intensity are investigated on both damage indicators. Theresults are derived by following the virtual testing schemeexplained in Subsection 3.1. The sample statistics of the damageindicators are obtained from the evaluation of 500 000 Latinhypercube samples, from which the probability density functioncan be estimated using kernel densities. By following thephilosophy of control charts, the false positive probability (theprobability of unobserved, but existing damage) is used as acriterion to assess the performance of the indicators. Basedon the undamaged structure, the control limits are defined atthe 2.5% and 97.5% quantiles, for Ip and m4 respectively. Thelimits are defined using the estimation of the inverse cumulativedistribution function based on kernel densities. Therefore, thenormality assumption for the indicators does not need to befulfilled. Nevertheless, as standard procedures assume often anormal distribution, the obtained distributions of the indicatorsare discussed in the light of the normality assumption.

4.3 Influence of time length

Based on the structural system described in previous subsec-tions, the probability density function estimates of the indicatorIp with respect to the time length and the load step are calculatedand shown in Figure 6a. It can be observed that for all loadsteps an increasing time length reduces the width (variance) ofthe probability density function, but keeps the position of themean value almost constant. The probability density functionsof the indicator m4 are depicted in Figure 6b. A change of thewidth and the position of the probability density function can be


2292

02468

1012

0.2 0.4 0.6 0.8 1 1.2 1.4

prob

abili

tyde

nsity

peak indicator Ip [-]

(a) time length influence on Ip

LS 15LS 50LS 60

0

100

200

300

400

0 0.005 0.01 0.015 0.02 0.025 0.03

prob

abili

tyde

nsity

peak indicator m4 [-]

(b) time length influence on m4

LS 15

LS 50

2s 4s 8s 16s

Figure 6: Influence of time length on the indicators’ probabilitydensity functions related to load steps (LS) 15, 50, and 60.

10−5

10−4

10−3

10−2

10−1

100

10 20 30 40 50 60fals

epo

sitiv

epr

obab

ility

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2s

4s

8s

16s

indicator Ip

indicator m4

Figure 7: Influence of time length on the false positiveprobability of the indicators.

observed with changing time length. With increasing damagethe variance and the mean value of the indicator increasessignificantly. Hence, the results of load step 60 are out of therange of Figure 6b.

Figure 7 illustrates the evolution of the false positive detectionprobability in dependency of the load step and the time length.With increasing time length both indicators become moresensitive to damage. A false positive probability of 10−2 orlower can be interpreted as sensitive to detect damage. Whilefor a time length of 2s the indicator m4 outperforms the indicatorIp, it is vice versa for a time length of 16s.

For both indicators, a large time length is beneficial for thedetection of damage.

4.4 Influence of measurement errors

The results of the probability density function estimates basedon the indicator Ip with increasing measurement error intensityν are presented in Figure 8a for a time length of 8s. Forthe load step 15 related to the undamaged structure, themeasurement error intensity has no influence on the probabilitydensity functions. If damage is present (see load step 50), theprobability density functions are notably different. The higherthe signal-to-noise ratio, the larger the mean value shift and the

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(a) measurement error influence on Ip

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0 0.01 0.02 0.03 0.04 0.05

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(b) measurement error influence on m4

LS 15 LS 50

ν = 0 ν = 1 ν = 2 ν = 4

Figure 8: Influence of measurement error intensity ν on theindicators’ probability density functions related to load steps 15and 50.

10−5

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epo

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ν=

4

ν=

2

ν=

1

ν=

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indicator Ip

indicator m4

Figure 9: Influence of measurement error intensity ν on thefalse positive probability of the indicators.

lower are the variances, which is beneficial for damage detectionas expected.

The probability density functions of the indicator m4 areshown in Figure 8b. For this indicator, the probability densityfunctions related to the undamaged and damaged structuredepend strongly on the measurement error intensity. The higherthe signal-to-noise ratio is, the larger the mean value and thelarger the variance of the indicators.

By assessing the false positive probability shown in Figure 9,it can be observed that the evolution of both indicators is almostidentical with respect to the variation of the measurement errorintensity.

4.5 Normality assumption

In practice, control charts with certain control limits are appliedto monitor damage indicators over time. Most of the controlcharts assume a normally distributed variable. Hence, it needsto be investigated, if the normality assumption is at leastapproximately fulfilled. This will be exemplarily illustrated bya representative configuration using a time length of 8s with ameasurement error intensity of ν = 1 for load step 50. Thehistogram, the probability density function estimation usingkernel densities, and the probability density function related to


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0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

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(b) distribution fitting m4

histogram kernel normal

Figure 10: Distribution fitting for a time history of length 8s, ameasurement error intensity of ν = 1, and the load step 50.

a fitted normal distribution are compared in Figure 10. It can beobserved that the distribution of the indicator Ip deviates slightlyfrom a normal distribution while the distribution of the indicatorm4 is significantly different from a normal distribution. Thisobservation is also valid for other combinations of time lengthand damage intensity. However, for both indicators, a large timelength and an undamaged structure lead to distributions veryclose to normal.

5 CONCLUSIONS

In this paper the variation of the statistical properties oftwo different damage indicators based on modal filter wereinvestigated within a virtual testing scheme with respect todamage intensity, measurement error intensity and time lengthof the considered vibration response time history. An implicitgradient damage law was applied to define realistic damagepattern for a simply supported beam with progressing damage.Assuming an ambient vibration experiment with knownstatistics of a multivariate normally distributed excitation, thestatistics of the damage indicators are derived in a two stepapproach. First, the statistics of the dynamic response Fouriertransforms of a modal filter was derived by using the approachproposed in [9][10]. This avoids the computationally verydemanding numerical time integration as typical needed instandard Monte Carlo methods. Secondly, a Latin hypercubesampling has been applied to derive the sample statistics of theindicators for each load step. This operation is computationallyvery inexpensive, which allows generating a large number of500 000 Latin hypercube samples.

Due to the different definitions of the indicators, the effects ofthe investigated influences on the probability density functionsare very dissimilar. Nevertheless, if the false positive probabilityis used as an assessment criterion, both indicators show asimilar performance to detect damage. It could be derivedthat the increase of the time length increases significantly theperformance of both indicators. With a large time length and a

typical signal-to-noise ratio, both indicators were able to detecta rather small damage.

The shape of the probability density functions has beendiscussed for both indicators with respect to damage intensityand time length. The normality assumption, as usually requiredfor the application of control charts in practice, is approximatelyfulfilled for the indicator Ip with a sufficiently large time length.For the indicator m4 the assumption of a normal distribution washardly justified for the investigated example, but the similaritycan be improved by increasing the time length.

Further research will be related to the investigation ofcorrelated excitations, as for example, typical for wind inducedexcitations.

REFERENCES[1] A. Deraemaeker and K. Worden, Eds., New trends in vibration based

structural health monitoring, vol. 520 of CISM Courses and Lectures,chapter Vibration based structural health monitoring using large sensorarrays: Overview of instrumentation and feature extraction based on modalfilters, pp. 19–54, SpringerWienNewYork, 2010.

[2] G. Tondreau and A. Deraemaeker, “Local modal filters for automated data-based damage localization using ambient vibrations,” Mechanical Systemsand Signal Processing, vol. 39, no. 1-2, pp. 162 – 180, 2013.

[3] G. Tondreau, Damage localization in civil engineering structures usingdynamic strain measurements, Ph.D. thesis, Universite libre de Bruxelles,Belgium, 2013.

[4] G. Tondreau and A. Deraemaeker, “Experimental localization of smalldamages using modal filters,” in Proceedings of IMAC XXXI A Conferenceand Exposition on Structural Dynamics, February 11 - 14, 2013.

[5] G. Tondreau and A. Deraemaeker, “Experimental automated localizationof small damages of a steel beam using local modal filters,” in Proceedingsof 9th International Conference on Structural Dynamics (EURODYN),June 30 - July 2, Porto, Portugal, 2014.

[6] J. Maeck and G. De Roeck, “Damage assessment using vibration analysison the Z24-bridge,” Mechanical Systems and Signal Processing, vol. 17,no. 1, pp. 133 – 142, 2003.

[7] A. Deraemaeker, “Assessment of damage localization based on spatialfilters using numerical crack propagation models,” in Proceedings of 9thInternational Conference on Damage Assessment of Structures (DAMAS),July 2011, Oxford, UK, 2011.

[8] T. E. Fricker, J. E. Oakley, N. D. Sims, and K. Worden, “Probabilisticuncertainty analysis of an FRF of a structure using a gaussian processemulator,” Mechanical Systems and Signal Processing, vol. 25, no. 8,pp. 2962–2975, 2011.

[9] M. Brehm and A. Deraemaeker, “An efficient method for the quantificationof the frequency domain statistical properties of short response timeseries of dynamic systems,” in Proceedings of IMAC-XXXII Conference& Exposition on Structural Dynamics, February 3-6, Orlando, USA,accepted, 2014.

[10] M. Brehm and A. Deraemaeker, “Uncertainty quantification of dynamicresponses in the frequency domain in the context of virtual testing,”Journal of Sound and Vibration, 2014, submitted.

[11] M. Brehm, T. J. Massart, and A. Deraemaeker, “Modelling of concretecracking for the design of SHM systems: comparison of implicit gradientdamage models and simplified linear representations,” in Proceedingsof 8th International Conference on Fracture Mechanics of Concrete andConcrete Structures (FramCoS), March 10 – 14, Toledo, Spain, 2013.

[12] M. Brehm, T. J. Massart, and A. Deraemaeker, “Towards a more realisticrepresentation of concrete cracking for the design of SHM systems:updating and uncertainty evaluation of implicit gradient cracking models,”in 6th European Workshop on Structural Health Monitoring, July 3 – 6,Dresden, Germany, 2012.

[13] H. G. Natke, Baudynamik, B.G. Teubner Stuttgart, 3. edition, 1989.[14] F. Zhang, Matrix Theory: Basic Results and Techniques, Springer, 2011.[15] H. G. Natke, Einfuhrung in die Theorie und Praxis der Zeitreihen- und

Modalanalyse, Vieweg & Sohn, 3. edition, 1992.[16] K. H. Jarman, D. S. Daly, K. K. Anderson, and K. L. Wahl, “A new

approach to automated peak detection,” Chemometrics and IntelligentLaboratory Systems, vol. 69, no. 1-2, pp. 61–76, 2003.

[17] H.R. Martin and F. Honarvar, “Application of statistical moments tobearing failure detection,” Applied Acoustics, 1995.

[18] R. H. J. Peerlings, Enhanced damage modelling for fracture andfatigue, Ph.D. thesis, Eindhoven University of Technology, Eindhoven,The Netherlands, 1999.


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