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Structural Engineering International

ISSN: 1016-8664 (Print) 1683-0350 (Online) Journal homepage: https://www.tandfonline.com/loi/tsei20

On Damage Detection System Information forStructural Systems

Sebastian Thöns, Michael Döhler & Lijia Long

To cite this article: Sebastian Thöns, Michael Döhler & Lijia Long (2018) On Damage DetectionSystem Information for Structural Systems, Structural Engineering International, 28:3, 255-268,DOI: 10.1080/10168664.2018.1459222

To link to this article: https://doi.org/10.1080/10168664.2018.1459222

Published online: 18 Jul 2018.

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On Damage Detection System Information for StructuralSystemsSebastian Thöns , Associate Professor, Department of Civil Engineering, Technical University of Denmark, Lyngby, Denmark and

Department 7 Safety of Structures, BAM Federal Institute for Materials Research and Testing, Berlin, Germany; Michael Döhler,

Research Scientist, Inria/IFSTTAR, Rennes, France; Lijia Long, PhD Student, Department 7 Safety of Structures, BAM Federal

Institute for Materials Research and Testing, Berlin, Germany. Contact: [email protected]

DOI: 10.1080/10168664.2018.1459222

Abstract

Damage detection systems (DDSs) provide information on the integrity ofstructural systems in contrast to local information from inspections or non-destructive testing (NDT) techniques. In this paper, an approach is developedthat utilizes DDS information to update structural system reliability andintegrate this information into risk and decision analyses. The approachincludes a novel performance modelling of DDSs accounting for the structuraland measurement system characteristics, the damage detection algorithm(DDA) precision including type I and II errors. This DDS performancemodelling provides the basis for DDS comparison and assessment inconjunction with the structural system performance including the damage andfailure state dependencies. For updating of the structural system reliability, anapproach is developed based on Bayesian updating facilitating the use of DDSinformation on structural system level and thus for a structural system riskanalysis. The structural system risk analysis encompasses the static, dynamic,deterioration, reliability and consequence models, which provide the basis forcalculating the direct risks due to component failure and the indirect risks dueto system failure. Two case studies with the developed approach demonstrate apotential risk reduction and a high Value of DDS Information.

Keywords: damage detection; value of information; structural systems; damagedetection uncertainty modelling, structural system updating.

Introduction

Damage detection systems (DDSs)provide information about the integ-rity and performance of structuralsystems. The performance of a struc-tural system is characterized by itssafety, i.e. the system reliability andrisks, and its functionality namely theassociated expected benefits andcosts. Structural system reliability isgenerally modelled with failure anddamage mechanisms that are basedon static and dynamic system behav-iour. These mechanisms generallyhave a statistical dependence causedby common influencing factors suchas environmental conditions, pro-duction and construction processesand material characteristics.1–3

Damage detection information ischaracterized by providing an indi-cation of damage of a structuralsystem subjected to a finite precision.4

The damage detection informationprovided by a measurement systemand a damage detection algorithm(DDA) refers to a change in the

structural system characteristics inrelation to a reference state of thestructural system.5,6 Damage detectioninformation thus accounts for thedependencies in the entire structuralsystem.

Both structural system performancemodels and DDAs have progressedsignificantly but separately in the scien-tific literature in recent decades.5,7–12

However, it has been demonstratedthat structural system identificationbased on response measurements canbe combined with the updating ofstructural reliability and that suchinformation can contribute to the accu-racy of the structural condition andthus to structural safety.13

This paper thus focuses on: (a) expli-citly modelling the characteristics of astructural system and its DDS, includ-ing the costs, consequences and func-tionalities, (b) utilizing the DDS forupdating the structural systemreliability, and (c) identifying the con-ditions under which DDS informationmay have a high value by utilizing

Bayesian decision analysis. The devel-oped approaches are elaborated withtwo case studies, namely (1) anexample focusing on how the structuralsystem reliability can be updated andhow a risk reduction can be achievedand (2) a case study containing aValue of DDS Information analysisof a bridge girder under highdegradation.

Structural PerformanceModelling, Inspection andDamage DetectionPerformance Modelling

Structural System Performance andRisk Modelling

Structural system reliability and riskare described by means of structuralsystem theory, which is based on theprobabilistic mechanical behaviour ofthe system in conjunction with itsloading, resistance and deteriorationmodels and the quantification of thesystem risks and utilities in conjunctionwith a consequence model.

Structural failure can be caused by anextreme static or dynamic loading ofthe system and/or its components. Anextreme structural system loadingimplies a high statistical dependenceof the component loadings. The depen-dence of the component resistances isgoverned by the production processes(e.g. section properties and materialparameters) and the constructionprocess (e.g. imperfections) of thestructure.1 Deterioration may causedamage to the structural system andcan affect the entire system or occurat different locations and in differenttime periods, depending on thematerial and the nature of thedamage mechanism. The spatial andcomponential dependence of damagemechanisms can thus vary significantly,such as for example in the fatigue ofsteel structures.14 The failure of a struc-tural system and/or its components

Structural Engineering International Nr. 3/2018 Scientific Paper 255

http://orcid.org/0000-0002-7714-3312mailto:[email protected]

caused by damage mechanisms thusrely on the structural system character-istics and the loading, resistance anddamage mechanism dependencies.9

The performance of a structural systemboth in regard to system failure andsystem damage can be described withlogical systems, Daniels systems andBayesian networks.7,15,16 In general,the probability P(FS) of a structuralsystem failure is calculated by integrat-ing the joint probability density overthe space of the system failure VFS :

P(FS) =∫VFS

fX(x)dX (1)

where the system failure space VFS isdefined with limit state functionsdependent on the vector of randomvariables X. The structural risks canthen be calculated with the nc com-ponent and system probabilities offailure (P(Fi) and P(FS)) and the con-sequences of component and systemfailure (Cc and CS). The risk may beclassified into direct risk due to com-ponent failure and indirect risk due tosystem failure (RD and RID) asfollows17:

R = RD + RID

=∑nci=1

P(Fi) · Cc + P(FS) · CS (2)

Damage Detection

Automatic damage detectionmethods in a structural health moni-toring (SHM) context are based onthe chosen measurement technologyfor the desired monitoring aim.4, 12

To illustrate the integration ofdamage detection information intothe performance analysis of a struc-ture, a vibration-based damagedetection method is considered inthe following.

Vibration monitoring is one of the bestknown and most well-developed tech-niques for long-term SHM and isrecognized as an addition or alterna-tive to visual inspections and themanual performance of local non-destructive testing (NDT).4,5,18,19 Therationale is that damage has an effecton the structural stiffness, and thus onthe modal parameters (modal frequen-cies, damping ratios and mode shapes)which characterize the dynamics of thestructure. A network of vibration

sensors (usually accelerometers) isattached to the structure to continu-ously measure the structural vibrationresponses to ambient excitation likewind, traffic, waves and other forces.Changes in the measured signals withrespect to the dynamic characteristicsof the structure then indicate thedamage that these forces might havecaused. Since changes in environ-mental conditions (such as tempera-ture fluctuations) also lead to changesin the signals, they must be accountedfor.20,21

Methods for vibration-based damagedetection compare the measurementdata from the (healthy) referencestate of the structure with data fromthe current, potentially damagedstate, and an alarm is raised if thedifference between both statesexceeds a threshold. There are manymethods for damage detection in thiscontext.6 For example, a straightfor-ward approach is to identify anddirectly compare the modal par-ameters from measurements of boththe reference and current states.18

Other methods indirectly comparecurrent measurements to a referencein a statistical difference measurewithout modal parameter identifi-cation. For instance, such methodsinclude non-parametric change detec-tion based on novelty detection,22

Kalman filter innovations,23 methodsfrom machine learning4 and manymore. Also belonging to this class ofmethods, statistical subspace-baseddamage detection methods offer a flex-ible and well-founded theoretical fra-mework that is used in thispaper.10,11,19,24,25

In the following, the subspace-basedDDA is introduced as an example ofa global automated damage detectionmethod. With this method, vibrationmeasurements from the currentsystem are compared to a referencestate in a subspace-based residualvector. In a hypothesis test, the uncer-tainties of the residual are taken intoaccount and the respective x2 test stat-istic is compared to a threshold in orderto decide whether or not the structureis damaged. Based on these properties,the x2 test statistic is considered as thedamage indicator value (DIV) fordamage monitoring.

Dynamic Structural System Model

The behaviour of the monitored struc-ture is assumed to be described by the

following linear time-invariant dyna-mical system:

Mz̈(t)+ Cż(t)+Kz(t) = vF(t) (3)

where t denotes continuous time, M, Cand K [ Rm×m are the mass, dampingand stiffness matrices, the vectorz [ Rm collects the displacements ofthe m degrees of freedom of the struc-ture and vF(t) is the external force thatis usually unmeasured for long-termmonitoring. Observing the systemdescribed by Eq. (3) with a set of racceleration sensors yields the follow-ing measurements:

y(t) = Lz̈(t)+ e(t) (4)

where y [ Rr is the measurementvector, the matrix L [ Rr×m indicatesthe sensor locations and e is themeasurement noise.

Measurements are taken at discretetime instants t = kt, where k is aninteger and τ is the time step. Asampling model consisting of Eq. (3)and Eq. (4) at the rate 1/τ and trans-formed into a first-order system yieldsthe following discrete-time state spacemodel:

xk+1 = Axk + vkyk = Cxk + wk

(5)

where the states, outputs, state tran-sition matrix and output matrix are asfollows:

xk =z(kt)

ż(kt)

[ ],

yk = y(kt),

A = exp 0 I−M−1K −M−1C

[ ]t

( ),

C = L −M−1K −M−1C[ ]

(6)

respectively and the appropriatestate noise and output noise termsare vk and wk, respectively. Thestate noise is related to the unmea-sured external force vF , while theoutput noise depends on both vFand the measured noise e. Bothnoise terms are assumed to bestationary white noise for the theor-etical outline of the damage detec-tion method. While this assumptionseems to be restrictive in real-world

256 Scientific Paper Structural Engineering International Nr. 3/2018

applications on civil structures wherethe ambient excitation may be mod-elled by non-stationary and colourednoise, some robustness of thedamage detection method to theseconditions has been shown,10 whichis supported by several case studieson structures in operation.19 Thetheoretical system order of thesystem in Eq. (5) is the dimensionof the states n = 2m. Note that theabove modelling is not only validfor acceleration measurements as inEq. (4) but generalizable to displace-ment and velocity measurements.

The modal parameters of the model inEq. (3) and Eq. (4)—namely thenatural frequencies, damping ratiosand observed mode shapes—areequivalently found in the systemmatrices (A, C) of the model in Eq.(5). Damage in the monitored systemwill correspond to changes in thematrices M, C and K in Eq. (3), forexample a loss of mass or loss of stiff-ness, and thus affect the systemmatrices (A, C) and the modal par-ameters. Hence, damage that occursin the model in Eq. (3) can equivalentlybe detected as changes in the modalproperties related to the systemmatrices (A, C) of Eq. (5).

Based on the measurements yk of themonitored system from a referenceand a possibly damaged state, aresidual vector is defined based on sub-space properties without identifyingthe system matrices (A, C), as outlinedin the following sections.

Subspace Properties

From the measurement data{yk}k=1,...,N , the correlationsRi = 1NSNk=1ykyTk−i are computed fori = 1, . . . , p+ q, where p and q arechosen parameters (usuallyp+ 1 = q) with min (pr, qr) ≥ n.25Then, they are filled into a blockHankel matrix:

H =

R1 R2 · · · RqR2 R3 · · · Rq+1... ..

. . .. ..

.

Rp+1 Rp+2 . . . Rp+q

⎛⎜⎜⎜⎝

⎞⎟⎟⎟⎠(7)

This matrix possesses the factorization

property H = O C, with the observabil-ity matrix:

O =CCA...

CAp+1

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎦ (8)

and the controllability matrix C.Matrix O can be obtained from asingular value decomposition (SVD)of H, truncated at the model order n,as:

H = UDVT

= (U1 U0)D1

D0

( )VT ,

O = U1D1/21

(9)

Note that the truncated singularvalues correspond to noise and areusually small, namely D0 ≈ 0.Once O is obtained from Eq. (7) andEq. (9), the system matrices (A, C)can be extracted from Eq. (8) for sub-space-based system identification,26,27

from which the modal parameters canbe obtained. Instead of carrying outthis system identification step,however, the subspace properties areused for the definition of a damagedetection residual and a subsequentdamage detection test.

Damage Detection Residual and Test

Let H0 be a Hankel matrix (Eq. (7))filled with data from a (healthy) refer-ence state, and let its SVD in Eq. (9)be given. Define its left null spacematrix S, obtained as S = U0 in Eq.(9), such that STH0 ≈ 0.Now, let the new measurements{yk}k=1,...,N from an unknown state ofthe system be given, from which anew Hankel matrix H is computed. Ifthe data comes from the system inthe reference state then S is still anull space of H since the modal par-ameters and thus the matrices (A, C)(up to a change of basis) areunchanged. Thus, STH ≈ 0 is a charac-teristic property of the system in thereference state.

However, if the system is damaged, themodal properties related to thematrices (A, C) and thus to O in Eq.(8) change, and the mean of theproduct STH deviates from 0. Notethat the matrix H is computed fromdata and does not depend on thestate basis of the matrices (A, C).

These properties lead to the definitionof the residual vector10,11,24:

z =��N

√vec (STH) (10)

whereN is the number of samples fromwhich H is computed. This residualvector is asymptotically Gaussian (fora large N) with a zero mean in thereference state and a non-zero meanin the damaged state.10 Thus, achange in the system corresponds to adeviation from 0 in the mean value ofthe residual vector. The correspondinghypothesis test (for a decision betweenH0: the system is in the reference stateand H1: the system is in the damagedstate) leads to the following test stat-istic:

d = zTS−1z (11)

where S = E(zzT) is the residualcovariance matrix. An estimate of Sis the sample covariance that is com-puted on several realizations of theresidual using measurement data inthe reference state. Due to the distri-bution properties of the residual, thetest statistic (Eq. (11)) is asymptoticallyx2 distributed with a non-centralityparameter in the damaged state.10,11

To decide whether or not the moni-tored structure is damaged, the teststatistic is compared to a threshold.Based on realizations of the test stat-istic in the reference state, thethreshold is typically defined toreduce the probability of false alarms(type I errors) occurring to below achosen level, while keeping in mindthat a lower probability of falsealarms (lower type I errors) also leadsto a lower probability of indicationsof small amounts of damage (highertype II errors) being detected. In prac-tice, a trade-off between both needs tobe made. In the following, the event ofindication is defined as the detection ofdamage, namely when the test statisticexceeds the threshold. Note that theterms probability of indication andprobability of detection can be usedequivalently. Thanks to the previousproperties, the test statistic (Eq. (11))is considered as the DIV in the follow-ing analysis.

Note that Eq. (11) is the non-parametricversionof thedamagedetection test.11,20

A detailed description of the statisticalframework and more theoretical andcomputational insight of this methodare given in Refs. [10, 11, 25].


Updating the StructuralSystem Reliability withDamage DetectionInformation

The approach to update the structuralsystem reliability and risks is presentedat the end of this section, based on anoutline of the inspection and measure-ment performance modelling and theDDS performance modelling detailedbelow.

Inspection and MeasurementPerformance Modelling

Inspection and measurement perform-ance can be modelled based on theapproaches of NDT, non-destructiveevaluation (NDE) and non-destructiveinspection (NDI) reliability model-ling.28–30 In the following, methods ofNDT modelling are reviewed and aconsistent performance model is intro-duced to provide the basis for the DDSperformance assessment.

The NDT reliability calculation isbased upon modelling or measuringthe signal given damage and the noise(given no damage) of the NDT hard-ware. Assume that an indicationevent I is defined when the signalexceeds a predefined threshold tD.Depending on this threshold, the con-ditional probabilities of the indicationevent are calculated with the signal sand the noise s0, where nD componentdamage states of a structural com-ponent are modelled as discrete anddisjoint, that is cj, j = 1, . . . , nD, andc0 is defined as the undamagedstate. The probability of the comp-lementary events indication (I) andno-indication (�I) for a componentdamage state cj, namely P(I|cj) andP(�I|cj), respectively, are determinedby integrating the probability densityf of the signal:

P(�I|cj) =∫tD−1

f (s|cj)ds (12)

P(I|cj) =∫1tD

f (s|cj)ds (13)

The probability of indication and no-indication given no damage (P(I|c0)and P(�I|c0)) are calculated by integrat-ing the noise s0:

P(�I|c0) =∫tD−1

f (s0|c0)ds0 (14)

P(I|c0) =∫1tD

f (s0|c0)ds0 (15)

Note that these probabilities are givenfor an upper boundary threshold tD,namely damage is indicated when thesignal exceeds this threshold, but theycan still easily be generalized to thecase of a lower boundary or two-sidedthresholds, for example. The probabilitydensities f of the noise and the signalcan be determined with inter-laboratorytests performed independently severaltimes (so-called round robin tests)which imply a frequentistic basis,28–30

or analytically and numerically by simu-lating the NDT process.31,32

With the above four equations, theprobability of indication, the prob-ability of a false alarm and the receiveroperating characteristics (ROCs) aredefined. An example of a probabilityof indication (or probability of detection)plot covering the probability of indi-cation given no damage, c0, and differ-ent damage states, cj, j = 1, . . . , nD, isdepicted in Fig. 1 for a constantthreshold. Please note that thediagram contains a (non-zero) prob-ability of indication given no damage(c0), which is also referred to as theprobability of false alarm.

The probability of a false alarm isdefined as the probability of indicationgiven an undamaged structural com-ponent P(I|c0).33 It is understood thatthe probability of a false alarm iscaused by noise, since damage isabsent by definition. Data normalizingefforts usually aim at reducing or aver-aging out the noise.12 The ROC is aplot of the probability of indication,P(I|ci), against the probability of afalse alarm, P(I|c0), for a particulardamage state ci but varying thethreshold tD.

The approach of NDT reliability mod-elling as outlined above has recentlybeen applied on a component level tomodelling DDA probabilities of indi-cation, for example with the purposeof assessing their quality.33–35 For thisaim the signal is defined as the DIVin the damaged state and the noise isdefined as the DIV produced by theDDA in the reference state, that isthe state where the structural systemis undamaged. With these definitions,the probabilities of indication and no-indication given no damage anddamage can be calculated as per Eq.(12), Eq. (13), Eq. (14) and Eq. (15).

Damage Detection SystemPerformance Modelling

In order to apply the above NDTreliability modelling to DDS, the fol-lowing important characteristics of theDDS information have to be takeninto account:

1. DDS information is provided byalgorithms processing measurementsystem signals.

2. The measurement system isattached to a structural system.

3. The DDS is operated by humans.

Regarding the first characteristic, theDIV is usually a random variable dueto the statistical signal processing offinite measurement data. Its statisticalproperties are influenced by variousfactors, including data length, measure-ment uncertainties and uncertainenvironmental conditions like theproperties of ambient excitation.Regarding the second characteristic,the DIV for damage detection is avalue that indicates changes in theentire structural system, meaning thatdamage in each of the componentscan have a different influence on theDIV. Hence, the damage states of thesystem need to be defined in conjunc-tion with the structural componentsand the structural system performance.DDS information is also subjected tohuman errors, as per the third charac-teristic, which need to be taken intoconsideration.31

These major DDS characteristics areincorporated as follows. The structuralsystem is discretized into nc com-ponents with discrete damage states.For example, these components maycorrespond to the elements of a finite-element method (FEM) model of thestructure. Let each of the componentsi [ {1, . . . , nc} have (nD + 1) possiblestates, namely an undamaged stateand nD damage states. For simplicityof notation, an equal number of nD

Fig. 1: Exemplary probability of indicationcurve for one component


damage states is assumed for eachcomponent, which can be easily gener-alized. The respective states aredenoted by ci,ji, where ji = 0 for theundamaged state and ji = 1, . . . , nDfor the damage states of component i.Then, a full discretization of the struc-tural system states contains the poss-ible combinations of all of thesestates, amounting to (nD + 1)nc − 1system damage states plus the unda-maged state. Each of these states thencorresponds to a particular set ofmatrices describing the monitoredsystem in Eq. (3), where the damagestates correspond to modifications ofthe matrices M, C and K. The systemdamage space Cnc is then defined bythe vectors cj = {c1,j1 , . . . , cnc ,jnc },where j = {j1, . . . , jnc }. Note thatc0 = {c1,0, . . . , cnc,0} denotes theundamaged state.

Let the DIV dj be given, which isobtained for any of these states cj,whether the undamaged state or adamage state, and let its probabilitydensity function be given by f (dj|cj).The probabilities of indication andno-indication given a damaged struc-tural system can then be calculated byintegrating the probability densities ofthe DIV, namely:

P(�IS|cj) =∫tS,D−1

∫f (dj|cj)ddj (16)

P(IS|cj) =∫1tS,D

f (dj|cj)ddj (17)

Note that the probability of no-indi-cation in Eq. (16) is a type II error. Inthe considered damage detectionmethod, the DIV is x2 distributed(Eq. (11)). Thus, the probabilitydensity function f corresponds to theclassical x2 distribution in the unda-maged state, and to a non-central x2

distribution in the damaged state. Anexample of the distributions of theDIVs d0 and dj with j = 0 are givenin Fig. 2. The threshold—which is setup from test values in the referencestate for a given type I error—isdepicted, along with the type II error

(corresponding to Eq. (16)) and theprobability of indication (correspond-ing to Eq. (17)).

The distribution of the DIV that isrequired to evaluate the probabilitiesin Eq. (16) and Eq. (17) may beobtained in two different ways. First,it can be derived from the theoreticalproperties of the DDA, the damagestate and (assumptions on the) statisti-cal properties of the measurementdata. For the presented damage detec-tion method, the distribution of theDIV is known depending on adamage parameterization.10, 36 In thiscase, the damage parameters can beobtained from an FEM model of thestructure. Second, the distribution ofthe DIV may be obtained numericallyfrom Monte Carlo simulations, wheremeasurement data is simulated in therespective damage states. This secondoption is particularly useful when thetheoretical DIV distribution isunknown or difficult to evaluate—however, it comes with an additionalcomputational burden.

An example of the probability of indi-cationcomputedwithEq. (17) isdepictedin Fig. 3 for a system with two com-ponents, building upon the subspace-based DDA presented in the previoussection. It is observed that theprobabilityof indication of a damage stateP(IS|{c1,j1 , c2,j2 }) with {j1 = 0, j2 = 0}is higher than for component damagestates (P(IS|{c1,j1 , c2,j2 }) with{j1 = 0, j2 = 0} or {j1 = 0, j2 = 0}).

Fig. 2: Scheme of probability density functions of the damage detection test statistic d in thereference and in a damaged state

Fig. 3: Example of a probability of indication curve for a system consisting of two com-ponents dependent on the component damage state c1,j1=1 . . .c1,j1=nD and c2,j2=1 . . .c2,j2=nDand cS,0 = {c1,0, c2,0}


Note that the determination of theprobability of indication for a struc-tural system accounting for all possibledamages states may be a non-trivialtask, as the number of damage statesincreases exponentially with anincreasing number of components.One approach to circumvent this chal-lenge is to infer the probability of indi-cation by the properties of the DDA.36

Structural System PerformanceUpdating

The approach to structural reliabilityupdating has undergone several devel-opments in recent decades. The earlyand comprehensive work in Ref. [37]contains a framework for structuralreliability updating that includesinspection and monitoring informationand accounts for measurement uncer-tainties. Recently, approaches haveemerged for modelling and updatingthe system deterioration state of struc-tures that take into account the aspectof spatial correlation among elementdeterioration.8,38–41 However, theimpact of deterioration on structuralsystem reliability is seldom includedin these works, and only very recentstudies can be found which integratedeterioration into structural systemreliability.42–44 These approaches havein common that only local informationprovided by inspections for example isutilized, that is these approaches do notcover the characteristics of the DDSinformation, namely the dependencyof the measurement system and struc-tural system and that DDS may beable to detect correlated componentdamage states with a higherprobability.

The formulation of the structural per-formance and the DDS performanceon a system level as presented abovefacilitates the updating of the structuralsystem reliability. The probability ofstructural system failure subjected tothe system damage Cnc(tS) given theDDS information of no-indication,P(FS(tS)|Cnc(tS)|�IS), can be deter-mined utilizing Bayesian updating forany point in time during the servicelife tS as follows:

P(FS(tS)|Cnc(tS)|�IS)

= P(�IS|FS(tS)|Cnc(tS))P(FS(tS)|Cnc(tS))

P(�IS|Cnc(tS))

= P(FS(tS)|Cnc(tS)> �IS|Cnc(tS))

P(�IS|Cnc(tS))(18)

It should be noted that the DDS infor-mation itself refers to a point or periodin time that is neglected here for clarity.

The probability of no-indicationP(�IS|Cnc(tS)) given the systemdamage state can be calculatedbased on the developed approachfor the DDS performance calculation(Eq. (17)). Following on from this,45

the marginal probability of no-indication is calculated by integratingthe product of the conditionalprobability of no-indication(1− P(IS|Cnc)) and the joint prob-ability density of the system damagespace fCnc (C

nc) over the damagestate spaces. The integration is per-formed over the space V�IS, which isdefined with the limit state functiong�IS as the difference between theprobability of indication P(IS|Cnc)and a uniformly distributed randomvariable u, holding:

P(�IS) =∫V�IS

(1− P(IS|Cnc))fCnc (Cnc)dCnc

(19)

with

V�IS = {g�IS = P(IS|Cnc)− u}

To illustrate the characteristics of DDSinformation, the risk-quantificationmodel is further detailed, which accountsfor the direct and indirect risks (RDDS andRDDSID ) at time tS with DDS informationand yields the following:

RDDS(tS) = RD(tS)+ RDDSID (tS)

=∑nci=1

P(Fi(tS)) · Cc

+ P(FS(tS)|�IS(tS)) · CS

(20)

Only the indirect risks are updateddue to the system characteristics ofthe DDS information. In contrast,the direct risks are updated by NDTinformation, as these refer to localindications of damage. It may beargued that DDS information mayapply to both the structural systemand its components. However, theupdating of the component perform-ance presupposes damage localiz-ation, which is outside the scope ofthis paper.

A risk reduction DR(tS) can be quanti-fied by subtracting the risk given by theDDS information and accounting forthe expected SHM costs E[CDDS(tS)]

from the total risk without DDS infor-mation, which leads to:

DR(tS) = R(tS)− (RDDS(tS)+ E[CDDS(tS)]) (21)

Relative risk reduction relates the riskreduction to the total risks not utilizingDDS information as follows:

DR(tS)R(tS)

=R(tS)− (RDDS(tS)+E[CDDS(tS)])

R(tS)

(22)

Example: Structural RiskReduction with DDS Informationfor a Simplistic Structural System

To illustrate the developed approach,the effects of the structural systemand DDS characteristics on structuralreliability and risk are quantified. Forclarity, a simplistic structural systemsubjected to deterioration consistingof two components and a DDS com-prising two sensors and a subspace-based DDA (Fig. 4) are described,with the system’s static, dynamic,deterioration, reliability and conse-quences characteristics defined asrequired and outlined in the previoussections.

The structural system properties aremodelled with distributed componentstiffness and mass subjected to a struc-tural damping of 2% for each mode(Table 1). The system behaviour is cal-culated using the FEM. For each of thestructural components, nD = 100damage states c1,j1 and c2,j2are con-sidered, corresponding to stiffnesslosses from 0 to 10%. The resulting

Fig. 4: Structural system with sensor locations


system damage states arecj = {c1,j1 , c2,j2 } for j1, j2 = 1, . . . , nD.Due to the absence of redundancy, thestructural system reliability is modelledas a series system, yielding:

P(FS)

= P⋃nc=2i=1

MR,iRi(tS)−MSSi ≤ 0( )

(23)

The formulation contains the numberof components nc = 2 with therandom variables component resist-ance Ri (dependent on the time tS)and the system loading S, along withtheir associated model uncertaintiesMR,i and MS, respectively. The time-dependent resistanceRi(tS) of the com-ponent i is modelled with the initialresistance Ri,0 and the time-dependentdamage Di(tS):

Ri(t) = Ri,0(1−Di(tS)) (24)

For clarity, the temporal dependenceof the damage is neglected in whatfollows.

The structural reliability model is sum-marized in Tables 2 and 3. The systemloading is represented with a Weibulldistributed random variable S, whichresults by equilibrium in the com-ponent loading Si. The loading andresistance model uncertainties and the

resistance model are determinedaccording to Ref. [1] as Lognormal dis-tributed with a standard deviation of10%. The component probability offailure is calibrated to 1 · 10−3 (whennot varied) by adjusting the mean ofthe component resistance in the unda-maged state based on ISO 2394(2015)2 and the JCSS ProbabilisticModel Code.1 The correlation of theresistances and the deterioration aremodelled with a coefficient of corre-lation of 0.5 when not varied.

The consequence model for the calcu-lation of risk builds upon the genericnormalized costs for componentfailure Cc = 1.0, the structural systemfailure CS = 100 (e.g. see Ref. [17])and the DDS, CDDS, comprising theDDS investment (1.33 × 10−4 perchannel), installation (1.33 × 10−4 perchannel) and operation (1.33 × 10−4

per year) in accordance with Ref. [46].

The DDS is modelled with the accel-eration sensors s1 in the x-directionand s2 in the y-direction, recordingthe responses yk from the system laidout in Eq. (3), Eq. 4 and Eq. (5) usingthe subspace-based DDA describedpreviously. Based on the dynamicstructural system model, 1000 datasets of length N = 10 000 at a samplingfrequency of 50 Hz are simulated foreach of the undamaged and damagedstates for both sensors from whitenoise excitation.

The probabilities of indication P(IS|cj)and P(IS|c0) are determined for theconsidered DIV in Eq. (11) with the

threshold tD corresponding to a 0.01probability of a type I error (see alsoFig. 2). The DDA takes into accountthe uncertainties related to themeasurement data of finite length N,which is due to the unknown ambientexcitation and the measurementnoise. Human errors in the applicationand operation are accounted for by themultiplication of the probability ofindication with a factor of 0.95.31

The probabilities of indication basedon data from sensors s1 ands2 dependent on the system damagestate defined as axial stiffnessreduction are depicted in Fig. 5. Itis observed that the probabilities ofindication based on sensor s2 are sig-nificantly higher, which is caused bythe higher axial stiffness of the systemin the y-direction and thus higherabsolute stiffness changes due to thesimulated damages. It is furtherobserved that damage of similar sizein both components can be detectedwith a higher probability thanindividual damage in one of thecomponents.

The DDS information is utilized toupdate the structural system reliabilityof the deteriorated structural system.The correlation characteristics of thedeteriorated structural system mayvary significantly due to the com-ponent, the system and the deterio-ration characteristics (as identifiedpreviously). The updated reliabilitiesand risks are thus depicted dependenton the resistance and deterioration cor-relation to allow for more generality ofthe example.

The deteriorated series system beforebeing updated by DDS informationshows a slight decrease in the structuralsystem failure probability for bothincreasing the deterioration and theresistance correlation rR0and rD(Fig. 6). When utilizing the DDS infor-mation, the failure probabilitydecreases and a higher decrease ratefor the deterioration correlation isobserved for sensors s1 and s2.However, both effects are significantlymore pronounced for sensor s2 due tothe higher probabilities of indication(Fig. 6). The higher decrease rate forthe damage correlation in comparisonto the resistance correlation isexplained by the higher probability ofindication for correlated instances ofdamage of the same size—namely thehigher the system deterioration, thebetter the DDS performance.

Parameter Value

Mass per component 0.5

Stiffness of component 1: EA1 1000

Stiffness of component 2: EA2 2000

Damping ratio 2%

Table 1: Structural model properties

Random variable Distribution* MeanStandarddeviation

Loading S WBL 3.50 0.10

Model uncertainty MS LN 1.00 0.10

Component resistance in undamagedstate R0,i

LN Calibrated 0.10

Model uncertainty MR,i LN 1.00 0.10

Damage Di N 0.07 0.03

* LN = Lognormal; N = Normal; WBL = Weibull.

Table 2: Structural reliability model

Randomvariables

Coefficient ofcorrelation

Resistance rR0 0.5 (when not varied)

Damage rD 0.5 (when not varied)

Table 3: Correlation model


An exponential dependency of thestructural system failure probabilityon the component probability offailure is observed (Fig. 6b). Thedependency is explained by the inter-dependency of the component failureand the system failure. The systemfailure probability is more reducedwhen using sensor s2 due to thehigher probability of indication ofstructural damage.

The risk reduction DR and the relativerisk reduction DR/R according to Eq.(21) and Eq. (22) dependent on thedamage and the resistance correlationare shown in Fig. 7. The risk reductionfor both sensors is positive despite con-sidering the expected SHM costs,which means that the expected SHM

costs are overcompensated for bythe risk reduction due to the DDSinformation.

It is observed that sensor s2 leads to asignificantly higher risk reductionthan sensor s1, which is caused by ahigher reduction in the probability ofsystem failure. The higher riskreduction decrease rate observed forthe damage correlation is in line withthe findings in Ref. [2].

The relative risk reduction relates theabsolute risk reduction to the totalrisks without utilizing the DDS infor-mation and can thus be seen as ameasure for the significance of therisk reduction. The risk reductionvaries between 2.3 and 6.0% for

sensor s1 and between 18.4 and32.8% for sensor s2 (Fig. 7b). Thebehaviour of the relative riskreduction dependent on the damageand the resistance correlation isvery similar to the risk reduction, asthe system failure probability varieslinearly and to a limited extent.

A higher absolute risk reduction isobserved for systems with a higherprobability of component and thussystem failure (Fig. 8a). This effecthas also been observed for a ductileDaniels system with an SHM strategyof load monitoring.44 However, therelative risk reduction (Fig. 8b)decreases, as the system risk increaserate is higher than the risk reductionrate.

Fig. 5: Probability of indication dependent on the system damage states for: (a) sensor s1; (b) sensor s2

Fig. 6: Prior and posterior system probability of failure (P(FS) and P(FS|�IS)) for different sensor positions dependent on the damage andresistance correlation (a) and the probability of component failure (b)


It has been demonstrated that a signifi-cantly higher uncertainty and riskreduction can be achieved with sensors2. The performance of the sensor inthe context of this study relies onmore efficiently capturing thedynamic behaviour of the structuralsystem, thus facilitating higher sensi-tivity and thus a higher probability ofdetecting small damages. For a furtheroptimization of DDS, these character-istics and the interrelations of the struc-tural and DDSs have to be considered.

Utility Gain Quantificationfor Damage DetectionInformation

The value of the damage detectioninformation, namely the utility gainby applying DDSs, can be quantifiedbased on the approach for the quantifi-cation of the value of the structural

health monitoring information2 (V) asthe difference between the life-cyclebenefits B1 and B0 with and withoutthe DDS strategy i:

V = B1 − B0 (25)

The expected value of the life-cyclebenefit B0 is formulated as aprior decision analysis, that is the max-imization of the expected benefitsb0 with the kn action choicesa = [a1 . . . ak . . . akn ]T and the ln struc-tural performance uncertaintiesXk = [X1 . . .Xl . . .Xln ]T :

B0 = EXk,l [b0(a∗,0k , Xl)]

with

a∗,0k = argmaxak

(EXk,l [b0(ak, Xl)]) (26)

Utilizing DDS strategies si, theexpected value of the life-cyclebenefit B1 is calculated by additionallyconsidering the jn uncertain DDS infor-mation Isi = [Isi,1 . . . Isi,j . . . Isi,jn ]T withthe extensive form of a pre-posteriordecision analysis:

B1 = EZsi ,j [E′′Xk,l [bi(s∗i , Isi,j, a∗,ik , Xl)]]

with

(s∗i , a∗,ik )=

argmaxsi

EIsi ,j [argmaxak

E′′Xl [bi(si, Isi ,j, ak,Xl)]]

(27)

The calculation of the expectedbenefits necessitates explicit benefit,cost and risk models dependent onDDS strategies, its outcomes, theactions and the life-cycle performance.These models are exemplarily

Fig. 7: Structural risk reduction dependent on the resistance and damage correlation for different sensor positions: (a) absolute risk reductionDR; (b) relative risk reduction DR/R

Fig. 8: Structural risk reduction dependent on the component probability of failure for different sensor positions: (a) absolute risk reductionDR; (b) relative risk reduction DR/R


developed in the next section for a casestudy with a bridge girder system thatis subjected to deterioration.

Case Study: Value of theDamage DetectionInformation for a Pratt TrussBridge Girder

A statically determinate Pratt trussbridge girder (Fig. 9) is analysed. It isassumed that the girder has been oper-ated for five years without inspection.A further operation until the end ofthe service life TSL of fifty years isintended. There are indications thatthe bridge may undergo an abnormaland very high deterioration. Thebridge manager knows that a DDSwill deliver more information on thecondition of the structure but doesnot know at which point in timeduring the service life the conditionassessment should be performed. Inorder to determine an optimal pointin time, the bridge manager performsan analysis of the value of the DDSinformation.

The bridge is assumed to be in twosystem states—namely in the failurestate or the safe state. Failure of thetruss is caused by component failuredue to non-redundant system charac-teristics and will lead to the costs CF.Component failure is caused byextreme loads in combination withthe damage development over time.The bridge manager has two options:do nothing (action a0) or repair(action a1), which will cost CR. Repairis modelled here as full reconditioningand it is assumed that all damagedcomponents are exchanged.

The probability of system failure P(FS)can be calculated with a series systemformulation with annual (year j)damage increments DD,i,j. The initialresistance is reduced by the square ofthe resistance reduction, which can beshown to be proportional to a

corrosion-induced diameter and conse-quently section loss. The section loss isalso directly proportional to the stiff-ness loss, for which the probability ofindication is determined.

P(FS)= P⋃nc=29i=1

MR,iRi,0

(

× 1−∑tj

DD,j

( )2−MSSi ≤ 0

⎞⎠

(28)

The static, dynamic and structuralreliability models of the Pratt trussbridge girder are summarized inTables 2 and 4. The annual deterio-ration has been assessed and DD,i,j isassumed to follow a Lognormal distri-bution with a mean of 0.001 and a stan-dard deviation of 0.001. The mean ofthe resistance Ri,0 is calibrated to aprobability of 1 · 10−6, disregardingany damage and considering that theconsequence of failure is large andthe relative cost of safety measures issmall.1,47 The probabilistic annualextreme loading S is applied verticallyon the truss and evenly distributed onthe lower nodes 2, 3, 4, 5, 6, 7 and 8with Si = 1/7 · S.The cumulative probability of systemfailure with time with and changes ofthe damage and resistance correlationcoefficient is shown in Fig. 10. For aconstant coefficient of correlation, theprobability of system failure increaseswith time due to the accumulateddeterioration damage. When varyingthe coefficient of correlation from0.1 through 0.5 to 0.9, the results indi-cate that with an increase in the coeffi-cient of correlation, the probability ofsystem failure decreases.

The DDS is modelled with accelerationsensors located at nodes 12, 13 and 14of the truss in the vertical direction torecord the response using the sub-space-based DDA (Fig. 11). The

probabilities of indication and theprobability of truss system failuregiven the DDS information of no-indi-cation are calculated following themethod laid out previously. The indi-vidual probabilities and an exemplaryjoint probability of indication areshown in Fig. 12. A DDS employmentcauses costs of CDDS and may causealso damage localization costs Cloc incase damage is indicated.

In Fig. 13, the effect of updating thestructural system reliability is shownfor one DDS utilization (see Eq. (18))for the different considered damagefailure correlations and for differentDDS employment years.

The decision analysis for the quantifi-cation of the value of the DDS infor-mation V is presented in Fig. 14,based on the formulation laid outearlier in the paper. The analysis

Fig. 9: Decision tree for assessing the value of DDS information (where the rectangles are decision nodes and the circles are chance nodes)

Parameter Value

Mass per component 0.02

Young’s modulus E 14400

Cross-section A 10/144

Length of non-diagonal element 10

Length of diagonal element 10��2

√

Damping ratio 2%

Table 4: Structural model properties

Fig. 10: Probability of system failure over timewith varied values of rR0,i = rD = 0.1, 0.5, 0.9


encompasses the basic decision ofwhether or not to employ a DDS forcondition assessment, the DSS strat-egies, their outcomes, the actions andthe system service life performance,as well as the associated consequencesfor each branch of the decision tree.

The cost model based onRefs. [48, 49] isshown in Table 5. All costs are dis-counted with a discount rate of 0.02.During the service life of the bridge, atarget system failure probability of1 · 10−4 is required, as the costs forsafety measures are high and the

consequences remain high.1, 47 Thecost of repair CR increases with timedue to the damage accumulation andis modelled dependent on the invest-ment cost CI and the service life TSL(Eq. (29)). When the bridge is repaired,it is assumed that all damaged

Fig. 12: Probabilities of: (a) indication dependent on the stiffness loss in 29 components; (b) joint probability of indication for components 1and 2

Fig. 13: Posterior probability of system failure: (a) for P(FS|�IS)with varied values of rR0,i = rDD,i = 0.1, 0.5, 0.9 if implementing DDS at year10; (b) during service life when rR0,i = rDD,i = 0.5 if implementing DDS at year 10, 20, 30 or 40

Fig. 11: Pratt truss bridge girder with a distributed load (arrows) and the sensor locations (double arrows)


components are exchanged with newcomponents:

CR = CITSL + 2− t (29)

The failure cost CF is assumed to behigher than the investment costs by a

factor of 100 due to indirect conse-quences. The localization cost Cloc andDDS application cost CDDS areassumed equal to 0.1, following forexample Ref. [48].

Figure 15 shows the expected costs andrisks and the value of the DDS infor-mation dependent on the DDS

employment year tDDS. The value of theDDS information is positive betweenyears 7 and 11. The maximum is at year7, with a significant relative value of theDDS information of 11%. The value ofthe DDS information is very low in year6, at which point the DDS informationdoes not influence the repair actions.Only DDS employment in year 7 willlead to changes in the repair actions andhence a reduction in the expectedrepair costs. The value of the DDS infor-mation decreases in the consecutiveyears, as the period for which the DDSinformation provides a risk reductionbecomes shorter. A drop in the value of

Fig. 14: Decision tree combining a prior decision analysis (branch with no DDS) and a pre-posterior decision analysis to calculated theoptimal expected life-cycle benefit B∗ for one point in time.

VariableDiscountrate r

Investment costCI

Failure costCF

Localization costCloc

DDS costCDDS

Value 0.02 10 1000 0.1 0.1

Table 5: Parameters in the cost and benefit analysis

Fig. 15: (a) Expected costs and risks; (b) value of DDS information dependent on the DDS employment year tDDS for rR0,i = rDD,i = 0.9


the DDS information occurs in year 13,as here the optimal action also becomesrepair for the branch without the DDSinformation, as enforced by the targetreliability of 10−4.

Conclusions

The introduced approach facilitates theupdating of the structural reliabilityand risks with DDS information atthe system level for both the DDSand the structural system. The DDSperformance modelling accounts forthe characteristics and dependenciesof the structural system states andencompasses the measurement system(number of sensors, sensor positions,precision of the system incorporatinghuman errors) and the employedDDAs. The structural system damagestates necessitate consistent modellingin terms of the static, dynamic anddeterioration characteristics in orderto derive the structural reliability andrisk, as well as the DDS performancemodels.

The introduced DDS performancemodelling facilitates the comparisonand assessment of various DDSs onthe basis of the probabilistic indicationcharacteristics at the DDS and struc-tural system levels with for examplethe system- and/or component-wiseand structural system specific prob-ability of indication. The calculationof the DDS performance maydemand high experimental or compu-tational resources, as the systemdamage state space increases exponen-tially with the number of components.Several strategies to overcome thischallenge have been discussed andoutlined.

The quantification of the value of theDDS information may serve as a basisfor DDS design. In this perspective, aDDS (i.e. the number of sensors, thesensor positions, the precision, theDDA) can be optimized to achieve amaximum expected life-cycle benefitfor a specific structural system or classof structural systems.

With two case studies, the potential ofthe approach has been demonstratedin terms of the significant value of theDDS information and the reductionof the structural system risk by utilizinga subspace-based DDA. The approachis generalizable to other DDAs, requir-ing the indication characteristics for a

discretization of the system damagestate space for a chosen DDA.

Acknowledgements

The COSTAction TU1402 on Quantifyingthe Value of Structural Health Monitoringis gratefully acknowledged for the inspiringdiscussions, contributions and workshopsacross the scientific and engineering disci-plines with researchers, industrial expertsand representatives of infrastructure oper-ators, owners and authorities.

ORCID

Sebastian Thöns http://orcid.org/0000-0002-7714-3312

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AbstractIntroductionStructural Performance Modelling, Inspection and Damage Detection Performance ModellingStructural System Performance and Risk ModellingDamage DetectionDynamic Structural System ModelSubspace PropertiesDamage Detection Residual and Test

Updating the Structural System Reliability with Damage Detection InformationInspection and Measurement Performance ModellingDamage Detection System Performance ModellingStructural System Performance UpdatingExample: Structural Risk Reduction with DDS Information for a Simplistic Structural System

Utility Gain Quantification for Damage Detection InformationCase Study: Value of the Damage Detection Information for a Pratt Truss Bridge GirderConclusionsAcknowledgementsORCIDReferences

new on damage detection system information for structural systems · 2019. 1. 23. · on damage...

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