polynomial chaos expansion models for the … · 1 polynomial chaos expansion models for the 2...

POLYNOMIAL CHAOS EXPANSION MODELS FOR THE1

MONITORING OF STRUCTURES UNDER OPERATIONAL2

VARIABILITY3

Minas D. Spiridonakos1, Not a Member, ASCE, Eleni N. Chatzi,1 Member, ASCE

and Bruno Sudret,1 Not a Member, ASCE

4

ABSTRACT5

This study overviews the development and implementation of a stochastic framework6

efficiently fusing operational response data with external influencing agents, namely environ-7

mental conditions, for representing structural behavior in its complete operational spectrum.8

The delivered methodology results in the formulation of a structural performance indicator9

serving as a precursor of potential damage or deterioration. The proposed approach relies on10

the combined use of (i) a polynomial chaos expansion method, able to account for structural11

variations tied to the variation of the acting agents; and (ii) an independent component anal-12

ysis algorithm, for extraction of salient behavioral features. The efficacy of the introduced13

method is demonstrated on field data from actual large-scale systems in both operational14

and damaged states.15

Keywords: condition indicator, externally acting agents, environmental conditions, damage16

detection, polynomial chaos expansion (PCE), independent component analysis (ICA).17

INTRODUCTION18

Vibration-based monitoring methods (Deraemaeker and Worden 2010; Ostachowicz and19

Güemes 2013) have long surfaced as a promising approach to non-destructive evaluation,20

and a highly researched subdomain of the structural health monitoring (SHM) field. Recent21

1ETH Zurich, Institute of Structural Engineering, Department of Civil, Environmental and Geomatic

Engineering, Stefano-Franscini-Platz 5, 8093 Zurich, Switzerland. E-mail: [email protected],

[email protected], [email protected]

1

mailto:[email protected]:[email protected]:[email protected]

advances in sensor technologies, with associated decrease in corresponding costs, have further22

revealed the potential of these methods to permanently escort the structure throughout its23

entire life-cycle, reporting significant information related to structural condition. Automa-24

tion, ability to work on a system level, adaptability for various types of structures, on-line25

or off-line implementation, are only few of the features contributing to the afore-mentioned26

concept. However, in deciphering the extracted information a number of shortcomings need27

be overcome. The latter pertains to the common lack of loading records, since the usual28

source of excitation is ambient loading, and secondly the susceptibility of these structures to29

a number of uncontrollable operational conditions, such as temperature, temperature gra-30

dients, humidity, wind speed, traffic intensity loading and others (Sohn 2007; Peeters et al.31

2001). Even though the former problem is adequately treated today by a number of accurate32

and robust output-only (operational) identification methods (Peeters and De Roeck 2001;33

Reynders 2012), the latter is still under research and in focus of a number of recent studies34

(Cross et al. 2013; Magalhães et al. 2012; Kullaa 2011a).35

The effect of operational, i.e., loading and environmental, conditions on the structural36

dynamics has been examined in various studies over the last 20 years. For example, Far-37

rar and coworkers (Farrar et al. 1997) report eigenfrequency differences for the Alamosa38

Canyon Bridge of approximately 5% over a 24 hour period due to temperature and spatial39

temperature gradients variation. Alampalli in his paper (Alampalli 2000) identifies natural40

frequencies differences of up to 50% for an abandoned bridge in Claverack (NY) and at-41

tributes this variation to the freezing of the bridge supports. Maximum differences varying42

from 14 to 18% were also calculated for the first four natural frequencies of the Z24 bridge43

monitored during a nine-months period for the SIMCES project (Peeters and De Roeck44

2001). Catbas et al. (Catbas et al. 2008) found that the structural reliability of a long-span45

truss bridge in USA was highly affected by the ambient temperature, while Cross et al.46

(Cross et al. 2013) investigated the effect of temperature, traffic loading and wind speed on47

the modal properties of the Tamar (UK) suspension bridge and identified differences up to48

2

almost 5%. Finally, Yuen and Kuok focused on the description of the relationship between49

natural frequencies and environmental factors (temperature and humidity) for tall buildings,50

while also utilized one-year measurement of a 22-storey reinforced concrete building (East51

Asia Hall) (Yuen and Kuok 2010).52

The approaches that have been followed in the literature for incorporating or discarding53

the operational variability from structural models may be primary classified into two cate-54

gories: i) output-only methods which aim at discarding the influence of operational factors55

solely from vibration response measurements and/or extracted features (e.g. modal prop-56

erties) (Kullaa 2011b; Reynders et al. 2014), and ii) input-output methods which aim at57

modelling the relationship between the measured vibration data and/or the extracted fea-58

tures with respect to measured operational conditions (Peeters and De Roeck 2001; Yuen59

and Kuok 2010). Borrowed from the machine learning literature, the terms unsupervised60

and supervised learning, respectively, have also been used to describe the aforementioned ap-61

proaches. In the former category methods such as the Principal Component Analysis (PCA)62

(Magalhães et al. 2012) or its nonlinear ramifications (kernel PCA (Reynders and De Roeck63

2015), Factor Analysis), neural networks, and others have been employed for solving the64

unsupervised learning problem by searching and discarding patterns in multivariate output65

features, which may reveal the influence of the unobserved input variables. The supervised66

learning alternative on the other hand follows a somewhat different approach, which typically67

involves formulation of a regression problem.68

The present study follows the second approach (Spiridonakos and Chatzi 2014; Spiridon-69

akos and Chatzi 2015), which delivers additional insight into the mechanism of variation of70

structural properties (outputs) and its dependence upon influencing agents (inputs) (Wenzel71

2009), which are often quantifiable via observations. A functional representation between in-72

puts and outputs is delivered, tracking the behavior of the structure under study for a wide73

range of operational conditions, even potentially unobserved ones through extrapolation.74

Information on the inputs may be extracted by deploying a small number of sensors track-75

3

ing externally acting agents, such as environmental (temperature, humidity, wave, wind) or76

traffic/train crossing loads, along with the array of vibration sensors monitoring structural77

response.78

The methodological framework proposed herein relies on the expansion of appropriately79

selected structural features, such as modal characteristics of the healthy structure estimated80

by means of operational modal analysis methods, onto a polynomial chaos basis (PCE)81

(Soize and Ghanem 2004; Berveiller and Lemaire 2006; Ghanem and D. 2003; Blatman and82

Sudret 2010) for the projection of these estimates on the probability space of the measured83

influencing agents. This process is carried out during a training phase resulting in a model,84

which is able to predict the evolution of structural response based on information of the85

input variables. Following the training phase, the PC-based estimated statistical properties,86

e.g. the prediction error, may the serve as condition indicators for the monitored system87

warning of irregular behavior or revealing evidence of a damaged/deteriorated system.88

In order to illustrate the workings of the method, the proposed framework is applied on89

a series of data from actual structural systems, including the Überführung Bärenbohlstrasse90

bridge in Zürich, as well as the benchmark SHM problem of the Z24-bridge in Switzerland91

(Peeters and De Roeck 2001; Reynders and De Roeck 2009). The results of these studies92

demonstrate a tool, which may reliably be employed into practice for condition monitoring93

of large-scale real world structures operating in a wide range of conditions.94

The remainder of the paper is structured as follows: In Section 2 the SHM framework95

based on PCE and ICA is introduced. The application case studies and the analysis results96

are presented in Section 3. Finally, the conclusions of this study are summarized in Section97

4.98

STRUCTURAL IDENTIFICATION FRAMEWORK99

The developed SHM framework aims at a global representation of the system, in the100

sense that it may accurately describe the dynamic properties of the structure for a range of101

operational conditions. The method employs conventional system identification methods in102

4

order to extract statistical characteristics (features) of the healthy structure. A polynomial103

chaos expansion (PCE) model (Berveiller and Lemaire 2006; Blatman and Sudret 2010)104

is then adopted for the representation of the extracted features on the space spanned by105

random variables describing the measured operational conditions. The extracted features106

could be either modal properties of the structure or even the parameters of an identified107

model.108

A schematic diagram of the training phase of the proposed framework is illustrated in109

Fig. 1.110

Polynomial Chaos Expansion (PCE)111

PCE is a tool employed for the propagation of uncertainty through dynamical systems,112

due to uncertainty relating to the system parameters (Wiener 1938; Ghanem and Spanos113

1997; Ghanem and D. 2003). The non-intrusive variant of the method is employed herein,114

relying on expansion of the random output variables on polynomial chaos basis functions,115

which are orthonormal with respect to the probability space of the system’s random inputs.116

Only the fundamental concepts of the employed tool are provided herein for the purpose of117

completeness. The interested reader is referred to (Blatman and Sudret 2010) for a deeper118

insight into the details of the method.119

Consider a system S comprising M random input parameters represented by indepen-120

dent random variables Ξ1, . . . ,ΞM , e.g. temperatures measured at different locations of the121

structure, gathered in a random vector Ξ with joint Probability Density Function (PDF)122

pΞ(ξ). Since the input variables are herein linked to measured quantities, it is reasonable to123

assume that information may be collected concerning their probability distribution.124

The system output, e.g. natural frequency estimates, denoted by Y = S(Ξ) will also be125

random. Assuming that Y has finite variance, it may be expressed as follows:126

Y = S(Ξ) =∑

d(j)∈NMθjφd(j)(Ξ) (1)127

5

where θj are unknown deterministic coefficients of projection, d is the vector of multi-indices128

describing the multivariate polynomial basis, and φd(j)(Ξ) are the polynomial basis (PC)129

functions, which are orthonormal to pΞ(ξ). The basis functions φd(j)(Ξ) may be constructed130

through tensor products of the corresponding univariate polynomials.131

Each univariate probability density function may be associated with a well-known fam-132

ily of orthogonal polynomials, serving for construction of the corresponding PC basis. For133

instance, the normal distribution is associated with Hermite polynomials while the uni-134

form distribution with Legendre (Table 1). A list of the most common probability density135

functions along with the corresponding orthogonal polynomials and the relations for their136

construction may be found in (Xiu and Karniadakis 2002).137

Obviously, the resulting series must be truncated to a finite number of terms for purposes138

of practicality, commonly resulting in selection of the multivariate polynomial basis with a139

total maximum degree |dj| =∑M

m=1 dj,m ≤ P for every j. In this case the dimensionality of140

the functional subspace is equal to141

p =(M + P )!

M !P !(2)142

whereM is the number of random variables and P the maximum basis degree. However, more143

compact representations may be achieved by dropping the terms which do not significantly144

contribute to the accuracy of the model, judged in terms of a predefined criterion (Blatman145

and Sudret 2010; Blatman and Sudret 2011).146

When truncating the infinite series of expansion of Eq. 1, the resulting PCE model is147

fully parameterized in terms of a finite number of deterministic coefficients of projection θj.148

The parameter vector θj may be estimated by solving Eq. 1 in a least squares sense. Toward149

this end, the data of the input-output data have to be employed.150

In visualizing the connection to the problem handled herein, consider the example a151

structural system subjected to ambient excitation which is susceptible to variations of the152

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environmental conditions, which would herein serve as the system inputs. A given output153

variable, such as the estimate of one of the system’s natural frequencies, could be expanded on154

a PC basis constructed to be orthogonal to the experimentally estimated PDF of temperature155

data. In this way, an indication is obtained regarding the sensitivity of the natural frequencies156

of the structure to the manner in which stochasticity, relating to temperature, propagates157

through the structural system.158

At this point, it needs to be noted that the application of the PCE method on real data159

is accompanied by a number of difficulties which have to be overcome. These are:160

1. the constraint of independence between the input variables161

2. the statistical characterization of the input variable data, that is the estimation of162

the underlying PDFs163

3. the selection of appropriate PC subspace164

A first necessary assumption obviously lies in that the considered input variables need be165

independent. However, operational condition data is normally acquired from a dense grid of166

sensors measuring operational conditions is normally installed in large structures, and thus167

a considerable percentile of the measured inputs is highly correlated. In order to extract168

only a small number of independent latent input variables, the Independent Component169

Analysis (ICA) algorithm is herein utilized in a first step for obtaining a set of independent170

transformed input variables out of the original dependent set (Hyvärinen and Oja 2000).171

The ICA algorithm is described in the next section.172

The second point pertaining to the fitting of PDFs on the measured input variable data173

may be addressed by various approaches, including i) approximation of the underlying PDFs174

by well-known parametric PDFs, ii) transformation to uniformly distributed data via use of175

nonparametric kernel estimates for the cumulative distribution function, and iii) implemen-176

tation of a data-driven approach, namely arbitrary PC (aPC) (Oladyshkin and Nowak 2012).177

Finally, to what the third point is concerned, the Least Angle Regression (LAR) method178

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proposed in (Blatman and Sudret 2011) is herein adopted for the selection of a sparse PC179

basis.180

Once these preliminary processing steps are performed, the training phase of the scheme181

is carried out for extracting the PCE model. The schematic diagram of the training phase182

of the proposed framework is illustrated in Fig. 1. Following this phase, the PCE-based esti-183

mated statistical properties of the features may be further exploited for extracting condition184

indicators as explained int he applications section.185

Independent Component Analysis (ICA)186

ICA is a source separation method which aims at estimating independent unobservable187

(latent) variables which are mixed into observed variables (Hyvärinen and Oja 2000). The188

key of the method lies in its search for non-Gaussian components, in contrast to principal189

component analysis and other second order methods, which rely on the covariance matrix190

of random variables. To this end, ICA combines a static linear mixture model with higher191

order statistics in order to identify unobservable variables. More specifically, considering192

n observed linear mixtures of n independent components {si, i = 1, . . . , n} the following193

relation is assumed (Hyvärinen and Oja 2000):194

xj = a(j,1)s1 + a(j,2)s2 + . . .+ a(j,n)sn, for j = 1, . . . , n (3)195

or by using vector-matrix notation196

x = As (4)197

with x designating the random observed variables vector, while198

A =

a1,1 . . . a(1,n)

.... . .

...

an,1 . . . a(n,n)

199denotes the mixing matrix, and s is the vector of unobserved independent variables. The200

latter quantities are unknown and have to be estimated by the random vector x and the also201

8

unknown mixing matrix A. The vector of the independent latent variables s may be readily202

obtained via the inverse of the mixing matrix A, herein denoted as W = A−1:203

s = Wx (5)204

The ICA focuses on deriving matrix A. As already noted, the key in estimating the ICA205

model is the non-Gaussianity of the unobserved sources. Therefore, ICA estimation algo-206

rithms are based on nonlinear optimization methods, which aim at the maximization of the207

non-Gaussianity of each one of the latent variables wTx, with w designating a column vector208

of matrix W . The measure of non-Gaussianity may be based on kurtosis, negentropy, and209

others (Hyvärinen and Oja 2000). The flowchart of a typical ICA algorithm implementation210

is provided in Fig. 2.211

Finally, it should be added that the characteristics of the ICA-based approach are par-212

ticularly attractive within the context of the proposed SHM framework. The advantage is213

twofold since PCE requires feeding of independent input variables and, additionally, only214

a reduced number of latent variables may be identified depending on the results of the215

eigenvalue decomposition of the ICA algorithm. Within such a context, ICA may serve for216

critically reducing the large bulk of data acquired via SHM systems by extracting only a217

reduced number of salient input and output features.218

APPLICATION CASE STUDIES219

Implementation on Systems under Operational Conditions220

As part of a project funded by FEDRO on “Structural Identification for Condition As-221

sessment of Swiss Bridges”, the Überführung Bärenbohlstrasse bridge has been chosen as a222

suitable test case due to its dimensions, static system, and ease of access (Fig. 3). According223

to existing documents, the bridge was designed in 1977 and construction was completed in224

1980. This implies that the bridge is currently at 34 years of age, therefore not extremely225

aged. The bridge was monitored from July 2013 till July 2014 through a permanent system226

9

acquiring hourly ambient vibration response and environmental condition measurements.227

More specifically, 18 acceleration sensors measuring ambient vibration responses (vertical228

axis), and two environmental condition sensors for measuring air temperature and humidity229

were installed. After the twelve-month period, more than 6500 datasets were collected. For230

these datasets, the estimation of the modal properties of the structure based on the NExT-231

ERA method was realized. The method’s hyper-parameters (state space model order equal232

to 30; 80 Markov parameters for the construction of the Hankel matrix) were kept constant233

for all datasets and they were selected based on initial modeling of a randomly selected234

training set of vibration response data.235

The first four natural frequencies estimated via the NExT-ERA method are selected as236

the output variables to be represented by the proposed methodology. These estimates are237

plotted versus time in Fig. 4. It may be observed that all four natural frequencies vary with238

time in three different patterns: i) almost linear increase from July 2013 till end of November239

2013, ii) high variability from end of November 2013 till end of December 2013, and iii) linear240

decrease from January 2014 till July 2014 when they seem to actually return to their July241

2013 values.242

A substantial part of this variability may be attributed to the seasonal variation of the243

environmental conditions. This dependency is clearly depicted in Fig. 5 where the natural244

frequency estimates are plotted versus temperature and humidity. A bilinear temperature -245

natural frequency relationship may be observed between negative and positive temperatures,246

while a more complex relationship seems to characterize the dependency of the identified247

natural frequencies and humidity.248

In order to apply the PCE approach, the independence of the random input variables,249

i.e., temperature and humidity, has to be checked. Looking at the scatter plot of Fig. 6a, it is250

evident that these are highly correlated (correlation close to 0.75). Therefore, the ICA has to251

be implemented before PCE is applied on the natural frequency estimates. The scatter plots252

of the corresponding ICA-based estimates of the independent random (latent) variables are253

10

shown in Fig. 6b, with the variables indicating a correlation close to 0. It should be noted254

that these transformed variables no longer correspond to temperature and humidity.255

The identified modal frequencies are then expanded onto the probability space of the256

latent input variables through PCE. Toward this end, the input set is transformed into uni-257

formly distributed variables via use of their non-parametrically estimated cumulative distri-258

bution functions. Consequently, a PC basis consisting of multivariate Legendre polynomials259

is calculated.260

The set of natural frequency estimates is divided into an training (estimation) and a261

validation set. The training set comprises the first 1250 values corresponding to the first262

five months of the monitoring period till early December 2013, while the rest of the values263

are used as validation set, that is 1114 values. The estimation set values are expanded on a264

PC basis consisting of multivariate Legendre polynomials of maximum order five, that is a265

total number of 21 basis functions, while it was observed that increasing the maximum order266

further did not significantly increased the accuracy of expansion.267

It is observed that the PCE-model estimates are capable of reliably reproducing the268

estimation set values, while more importantly very good accuracy is achieved for values269

lying in the validation set, including the more challenging winter months of the monitored270

interval (Fig. 7).271

Nonetheless, damage detection still comprises a difficult decision, which has to be based272

on multivariate statistics relating to the PCE expansion errors for the four natural frequency273

estimates. In order to simplify this problem, the ICA algorithm is utilized for estimating274

a reduced number of independent random variables from the estimated natural frequencies275

(output variables). As a result, a single damage (or condition) indicator is extracted relying276

on a univariate statistical hypothesis testing for the characterization of the “health” state277

of the structure. By implementing the ICA once again, this time retaining a single latent278

output variable, the results shown in Fig. 8 are attained, corresponding to a one-dimensional279

damage indicator. Based on the statistical characterization of the PCE model prediction280

11

errors, confidence intervals may be calculated with respect to this simplified metric, and281

used for damage detection as described before.282

The strength of the methodology presented herein therefore lies in that it is able to283

account for the influence of exogenous factors, such as environmental or other operational284

conditions, upon structural behavior without use of a detailed numerical structural model.285

Instead, this is a tool easily generalizable for various types of systems, without necessitating286

a-priori knowledge on the structure itself, but merely a training interval during which the287

structure is monitored. Hence, such a process is only implementable within a long-term288

monitoring perspective.289

Implementation on Systems under Damaged Conditions290

The workings of the method will be now demonstrated via implementation on an ac-291

tually damaged case, namely the well known case study of the Z24 bridge (Switzerland),292

which was monitored for nine months and finally artificially damaged for the purposes of293

the BRITE-EURAM project SIMCES. The bridge, dated from 1963, was overpassing the294

A1 highway (Zurich-Bern), connecting Utzemstorf and Koppigen and its demolition was295

planned after the monitoring period since a new railway adjacent to the highway required a296

new bridge with a larger side span (Reynders and De Roeck 2009). The bridge was moni-297

tored from November 1977 till August 1998 through a permanent system acquiring hourly298

ambient vibration response and environmental condition measurements. More specifically,299

16 acceleration sensors measuring ambient vibration responses (13 of them are shown in Fig.300

9 while three more were installed on one of the piers), while 49 environmental condition301

sensors were installed for measuring air temperature, wind characteristics, humidity, bridge302

expansion, soil temperatures at the boundaries and bridge concrete temperatures (Reynders303

and De Roeck 2009).304

After the nine-month period, progressive damages were artificially induced within a pe-305

riod of a (August 10 - September 11, 1998). The monitoring system was still in operation306

while during night vibration tests based on hammer, dropping mass device and shakers were307

12

performed. The specific types of induced damages are summarized in Table 2, while more308

details on the damage scenarios and description of the simulation of the real damage cause309

may be found in (Reynders and De Roeck 2009).310

Vibration and environmental data311

In this study, the temperatures measured at six locations at the center of the mid-312

dle span along with the air temperature are used (Fig. 10), while the first four natural313

frequencies estimated through Stochastic Subspace Identification (SSI) and an automated314

operational analysis method are considered as the output of the structural system (Reyn-315

ders and De Roeck 2009). The initial data set of 5652 values, corresponding to the hourly316

obtained data of the monitoring system, are initially truncated to a 3932 long dataset for317

which estimates of all four natural frequencies are available (Peeters and De Roeck 2001).318

The dependency of natural frequency estimates on the temperature measured at the cen-319

ter of the web is shown in Fig. 12. A bilinear temperature/natural frequency relationship320

may be observed between negative and positive temperatures, while values depicting abnor-321

mal behavior, that may be attributed to damage, are clearly indicated only for the second322

mode.323

Even though the list of input variables affecting the structural behavior of Z24 is not324

limited to this set of temperatures, it is assumed that they may be adequate for describing a325

large percentage of the variability of the estimated modal properties (Peeters and De Roeck326

2001).327

Results328

As a first step, and before expanding the natural frequency estimates to the PC functional329

basis orthogonal to the PDFs of the random input variables, i.e. recorded temperatures, the330

independency of the latter has to be checked. Looking at the scatter plots of Fig. 11,331

drawn for each pair of measured temperatures, it is evident that they are highly correlated332

(correlation larger than 0.95 for almost all pairs), while the smallest correlation is 0.89.333

Therefore, ICA has to be implemented before PCE is applied on the natural frequency334

13

estimates. The scatter plots of the corresponding ICA-based estimates of the independent335

random (latent) variables, are shown in Fig. 13, with most of the variables indicating a336

correlation smaller than 0.1, with the larger correlation being 0.13. It should be noted that337

these transformed variables do not correspond to temperature anymore.338

The identified modal frequencies are then expanded as a function of the random latent339

input variables through PCE. To this end, the input variables are transformed into uniformly340

distributed variables by using their non-parametrically estimated cumulative distribution341

functions and PC basis consisting of multivariate Legendre polynomials is finally constructed.342

The set of natural frequency estimates is in this case as well divided into a training and343

validation set. The training set comprises 1500 values randomly selected from the first eight344

months of the monitoring period (one month prior to the first damage), while the remaining345

values are employed as a validation set. The latter includes the complete set recorded during346

the last month of the monitoring period, plus the complete set of values corresponding to347

the period of induced damages, i.e., 2744 values.348

The values of the estimation set are expanded onto a PC basis consisting of multivariate349

Legendre polynomials of maximum order three, that is a total number of 120 basis functions,350

while it is observed that increasing the maximum order further does not significantly increase351

the accuracy of the expansion. The standard deviations of the expansion errors, computed352

on the validation set, are summarized in Table 3, with the maximum being approximately353

equal to 0.13 Hz.354

It may be observed that the PCE-model estimates are capable of reliably reproducing355

the estimation set values, while more importantly a very good accuracy is also achieved for356

values belonging to the validation set for the interval prior to damage (Fig. 14). Naturally,357

predictions deviate for the period after the first damage is induced (August 9, 1998) which358

correctly translates to damage indication.359

In once again extracting a reduces order condition indicator, the ICA algorithm is uti-360

lized for estimating a reduced number of independent salient features from both the vector361

14

of temperature measurements (inputs) and the estimated natural frequencies (outputs). By362

inspecting the eigenvalues of the covariance matrices (second step of the ICA; Fig. 2) it is363

obvious that a single input latent variable is enough to describe the variance of the tempera-364

ture measurements in a percentage equal to 95% (Fig. 15a). At the same time, by applying365

the ICA on the vector of natural frequencies, and inspecting the eigenvalues of the corre-366

sponding covariance matrix (Fig. 15b), a single output latent variable is retained, which is367

able to account for almost 90% of the output variance.368

The PCE of the output to the input (latent variables), with just 4 basis functions (uni-369

variate Legendre polynomial of orders 0 to 3), renders the results demonstrated in Fig. 16,370

for which an abnormal trend is noted during the damage period clearly signaling the damage371

initiation and progression. As a next step to this framework, damage localization is fore-372

seen, which may be explored via exploring spatial correlations in a grid of sensors. Kriging373

methods could assist to such an end, and are left for a subsequent exploration.374

Finalizing, it must be noted that in previous work, related to the same benchamark375

problem (Peeters and De Roeck 2001), an ARX model was used to describe the temperature-376

modal frequency relationship. Similar damage detection results were obtained, while the377

description of nonlinearity was alleviated via use of the temperature range above zero.378

CONCLUSIONS379

Environmental condition data which are nowadays available in modern SHM systems380

of large-scale civil engineering structures should be accounted for in favor of identifying381

comprehensive dynamic models. For this reason, the present study proposes an identifica-382

tion framework which utilizes polynomial chaos expansions for the accurate description of383

the propagation of stochasticity stemming from environmental factors through the struc-384

tural response. In order to compress/reduce the amount of recorded environmental data385

and simultaneously identify a small number of statistically independent input variables, the386

independent component analysis method is proposed as a suitable tool. In addition, its appli-387

cability for extraction of simplified, or low dimensional damage indicators is also examined.388

15

The implementation of the method on two field case studies demonstrated the effectiveness389

and applicability of the proposed framework toward both aforementioned directions.390

ACKNOWLEDGMENTS391

The authors would like to gratefully acknowledge Prof. E. Reynders from the Department392

of Civil Engineering, KU Leuven, for providing the SIMCES project data, serving as the393

second test case discussed herein. This research was supported by the Federal Roads Office394

(FEDRO) of Switzerland under Project #AGB 2012/015.395

16

APPENDIX I. REFERENCES396

Alampalli, S. (2000). “Effects of testing, analysis, damage, and environment on modal pa-397

rameters.” Mechanical Systems and Signal Processing, 14(1), 63–74.398

Berveiller, M., S. B. and Lemaire, M. (2006). “Stochastic finite elements: a non intrusive399

approach by regression.” Eur. J. Comput. Mech., 15(1-3), 81–92.400

Blatman, G. and Sudret, B. (2010). “An adaptive algorithm to build up sparse polynomial401

chaos expansions for stochastic finite element analysis.” Probabilistic Engineering Mechan-402

ics, 25(2), 183–197.403

Blatman, G. and Sudret, B. (2011). “Adaptive sparse polynomial chaos expansion based on404

least angle regression.” Journal of Computational Physics, 230(6), 2345–2367.405

Catbas, F. N., Susoy, M., and Frangopol, D. M. (2008). “Structural health monitoring and406

reliability estimation: Long span truss bridge application with environmental monitoring407

data.” Engineering Structures, 30(9), 2347–2359.408

Cross, E., Koo, K., Brownjohn, J., and Worden, K. (2013). “Long-term monitoring and data409

analysis of the tamar bridge.” Mechanical Systems and Signal Processing, 35(1-2), 16–34.410

A. Deraemaeker and K. Worden, eds. (2010). New Trends in Vibration Based Structural411

Health Monitoring, Vol. 520 of CISM Courses and Lectures. Springer Vienna.412

Farrar, C. R., Doebling, S. W., Cornwell, P. J., and Straser, E. G.413

(1997). “Variability of modal parameters measured on the alam-414

osa canyon bridge.” 15th International Modal Analysis Conference415

(IMAC XV), .417

Ghanem, R. and Spanos, P. (1997). “Spectral techniques for stochastic finite elements.”418

Archives of Computational Methods in Engineering, 4(1), 63–100.419

Ghanem, R. G. and D., S. P. (2003). Stochastic Finite Elements: A Spectral Approach.420

Courier Corporation.421

Hyvärinen, A. and Oja, E. (2000). “Independent component analysis: algorithms and appli-422

17

cations.” Neural Networks, 13(4-5), 411–430.423

Kullaa, J. (2011a). “Distinguishing between sensor fault, structural damage, and environ-424

mental or operational effects in structural health monitoring.” Mechanical Systems and425

Signal Processing, 25(8), 2976–2989.426

Kullaa, J. (2011b). Vibration-Based Structural Health Monitoring Under Variable Environ-427

mental or Operational Conditions, Vol. 520 of CISM Courses and Lectures. Springer, Chap-428

ter 4, 107–181.429

Magalhães, F., Cunha, A., and Caetano, E. (2012). “Vibration based structural health moni-430

toring of an arch bridge: From automated oma to damage detection.” Mechanical Systems431

and Signal Processing, 28, 212–228.432

Oladyshkin, S. and Nowak, W. (2012). “Data-driven uncertainty quantification using the433

arbitrary polynomial chaos expansion.” Reliability Engineering & System Safety, 106, 179–434

190.435

W. Ostachowicz and J. Güemes, eds. (2013). New Trends in Structural Health Monitoring,436

Vol. 542 of CISM International Centre for Mechanical Sciences. Springer Vienna.437

Peeters, B. and De Roeck, G. (2001). “One-year monitoring of the z24-bridge: environmental438

effects versus damage events.” Earthquake Engineering and Structural Dynamics, 30(2),439

149–171.440

Peeters, B., Maeck, J., and De Roeck, G. (2001). “Vibration-based damage detection in civil441

engineering: excitation sources and temperature effects.” Smart Materials and Structures,442

10(3), 518–527.443

Reynders, E. (2012). “System identification methods for (operational) modal analysis: Re-444

view and comparison.” Archives of Computational Methods in Engineering, 19(1), 51–124.445

Reynders, E. and De Roeck, G. (2009). “Continuous vibration monitoring and progres-446

sive damage testing on the z24 bridge.” Encyclopedia of Structural Health Monitoring, C.447

Boller, F.-K. Chang, and Y. Fujino, eds., John Wiley & Sons, Chapter 126, 1–10.448

Reynders, E. and De Roeck, G. (2015). “Vibration-based damage identification: The z24449

18

bridge benchmark.” Encyclopedia of Earthquake Engineering, M. Beer, I. A. Kougioumt-450

zoglou, E. Patelli, and I. S.-K. Au, eds., Springer Berlin Heidelberg, 1–8.451

Reynders, E., Wursten, G., and De Roeck, G. (2014). “Output-only structural health moni-452

toring in changing environmental conditions by means of nonlinear system identification.”453

Structural Health Monitoring, 13(1), 82–93.454

Sohn, H. (2007). “Effects of environmental and operational variability on structural health455

monitoring.” Philosophical Transactions of the Royal Society A, 365(1851), 539–560.456

Soize, C. and Ghanem, R. (2004). “Physical systems with random uncertainties: Chaos rep-457

resentations with arbitrary probability measure.” SIAM Journal on Scientific Computing,458

26(2), 395–410.459

Spiridonakos, M. and Chatzi, E. (2014). “Polynomial chaos expansion models for shm under460

environmental variability.” Proceedings of Eurodyn 2014, 9th International Conference on461

Structural Dynamics, June 30 July 2, 2014, Porto, Portugal.462

Spiridonakos, M. and Chatzi, E. (2015). “Stochastic structural identification from vibrational463

and environmental data.” In: Beer M., Patelli E., Kougioumtzoglou I., Au I. (Ed.) En-464

cyclopedia of Earthquake Engineering: SpringerReference (www.springerreference.com).465

Springer-Verlag Berlin Heidelberg, 1–16.466

Wenzel, H. (2009). “The influence of environmental factors.” Encyclopedia of Structural467

Health Monitoring, Y. F. C. Boller, F.-K. Chang, ed., John Wiley & Sons, Ltd, Chapter468

120, 1–9.469

Wiener, N. (1938). “The homogeneous chaos.” American Journal of Mathematics, 60(4), pp.470

897–936.471

Xiu, D. and Karniadakis, G. E. (2002). “The wiener-askey polynomial chaos for stochastic472

differential equations.” SIAM Journal on Scientific Computing, 24(2), 619–644.473

Yuen, K. and Kuok, S. (2010). “Ambient interference in long-term monitoring of buildings.”474

Engineering Structures, 32(8), 2379–2386.475

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List of Tables476

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TABLE 1. Types of Wiener-Askey polynomial chaos and their underlying random vari-ables.

PDF Support Polynomials

Normal (Gaussian) (−∞,∞) HermiteGamma [0,∞) LaguerreBeta [0, 1] Jacobi

Uniform [−1, 1] Legendre

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TABLE 2. Types of artificial damages induced on bridge Z24 (Reynders and De Roeck2009).

# Date (1998) Damage type

1 4.8 First reference measurement

2 9.8 Second reference measurement

3 10.8 Lowering of pier, 20 mm




7 19.8 Tilt of foundation

8 20.8 Third reference measurement

9 25.8 Spalling of concrete, 24 m2

10 26.8 Spalling of concrete, 12 m2

11 27.8 Landslide at abutment

12 31.8 Failure of concrete hinge

13 2.9 Failure of anchor heads I

14 3.9 Failure of anchor heads II

15 7.9 Rupture of tendons I

16 8.9 Rupture of tendons II

17 9.9 Rupture of tendons III

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TABLE 3. Errors of the polynomial chaos expansion .

Natural freq. (Hz) std(PCE error) (Hz)

ω1 0.0266

ω2 0.0526

ω3 0.1001

ω4 0.1346

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List of Figures477

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FIG. 1. Schematic diagram of the proposed identification method.

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FIG. 2. Flowchart of the ICA algorithm.

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FIG. 3. Test-case: Überführung Bärenbohlstrasse bridge.

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FIG. 4. The first four identified natural frequencies of the bridge for the variousdatasets of the long-term monitoring system.

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FIG. 5. The first four identified natural frequencies versus (a) temperature; (b) hu-midity.

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FIG. 6. Histograms of (a) the measured temperature and humidity; (b) the latentvariables occurring after application of the ICA, along with corresponding scatter plots.

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FIG. 7. PCE natural frequency estimates for both the estimation and validation setcontrasted to the true NExT-ERA based estimates.

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FIG. 8. Extracted feature variable compared to the PCE modeling estimates (top) andthe extracted damage index based on the PCE prediction errors for the ÜberführungBärenbohlstrasse.

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FIG. 9. The Z24-Bridge: longitudinal section and top view (Reynders and De Roeck2009).

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FIG. 10. Cross-section of the Z24-Bridge and location of the thermocouples (Reyndersand De Roeck 2009).

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FIG. 11. The first four natural frequency estimates obtained through the SSI methodplotted against the temperature measured at the center of the web.

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FIG. 12. Histograms of the measured temperatures (upper row and left column) alongwith the scatter plots and the estimated correlation values.

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FIG. 13. Histograms of the ICA-based estimated independent random variables (upperrow and left column) along with the scatter plots and the estimated correlation values.

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FIG. 14. PCE natural frequency estimates for both the training and validation setcontrasted to the true SSI-based estimates.

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FIG. 15. Normalized eigenvalues of a) the temperature vector and b) natural frequencyestimates vector.

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FIG. 16. PCE estimates for both the estimation and validation set based on a singleinput and a single output variable.

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Polynomial Chaos Expansion (PCE)Independent Component Analysis (ICA)Implementation on Systems under Operational ConditionsImplementation on Systems under Damaged ConditionsVibration and environmental dataResults

polynomial chaos expansion models for the … · 1 polynomial chaos expansion models for the 2...

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