Probabilistic Identification of Simulated Damage on the Dowling Hall Footbridge through Bayesian FE Model Updating

Transcription

1 Probabilistic Identification of Siulated Daage on the Dowling Hall Footbridge through Bayesian FE Model Updating Ian Behanesh and Babak Moaveni Dept. of Civil and Environental Eng., Tufts University, Medford, MA, USA ABSTRACT This paper presents a probabilistic daage identification study on a full-scale structure, the Dowling Hall Footbridge, through a Bayesian finite eleent (FE) odel updating. The footbridge is located at Tufts University and is equipped with a continuous onitoring syste that easures its abient acceleration response. A set of data is recorded once every hour or when triggered by large vibrations. The odal paraeters of the footbridge are extracted fro each set of data and are used for FE odel updating. In this study, effects of physical daage are siulated by loading a sall segent of the footbridge deck with concrete blocks. The footbridge deck is divided into five segents in a FE odel of the test structure and the added ass on each segent is considered as an updating paraeter. Overall, 7 sets of data are collected during the loading period and different subsets of these data are used to find the location and extent of the daage (added ass). The Adaptive Metropolis-Hastings algorith with adaption on the proposal probability density function is successfully used to generate Markov Chains for sapling the posterior probability distributions of the five updating paraeters. Effects of the nuber of data sets used in the identification process are investigated on the posterior probability distributions of the updating paraeters. The probabilistic odel updating fraework accurately predicts the siulated daage and the level of confidence on the obtained results. The probabilistic daage identification results are found to be in good agreeent with their corresponding deterinistic counterparts. KEYWORDS: Bayesian FE odel updating, Adaptive Metropolis-Hastings algorith, daage identification, uncertainty analysis, Dowling Hall Footbridge. INTRODUCTION Recent structural failures have raised public attention on the need for iproved infrastructure safety and aintenance [-]. Bridge and infrastructure owners are encouraged to evaluate and assess the true state of structural health regularly. This is currently done ostly by visual inspection. Structural health onitoring (SHM) can provide ore revealing and quantitative inforation on structural health and is copleentary to visual inspection. The ultiate goal of SHM is to deterine the four levels of daage identification proposed by Rytter [3] at the earliest possible stage: () existence of daage; () location of daage; (3) severity of daage; and (4) reaining useful life of structures. Many ethods have been proposed in the past two decades to address soe or all the daage identification levels [4-6]. A coonly used class of

2 these ethods is the finite eleent (FE) odel updating ethods [7-], which have been successfully applied for daage identification of civil structures in recent years. In the FE odel updating ethods, odel paraeters are adjusted so that odel predicted tie histories, odal paraeters, or frequency response functions best atch the corresponding quantities obtained fro the test data. However, the accuracy and spatial resolution of deterinistic daage identification results depend on the accuracy and copleteness of the easured data or the features extracted fro data (e.g., odal paraeters) and on the accuracy of the FE odel. The extracted data features are usually estiated with significant uncertainties, ainly due to easureent noise, estiation errors, and changing environental conditions. These uncertainties propagate through the FE odel updating process yielding uncertain odel updating results [-5]. Another unavoidable source of uncertainty in the odel updating process arises fro odeling errors, which are due to the siplifications and idealizations involved in the odeling. In order to better quantify the accuracy of odel updating results, it is necessary to account for these sources of uncertainties through a probabilistic odel updating fraework, where the prior probabilities of odel paraeters are updated to their posterior probabilities using the easured data. The papers by Sohn and Law [6] and Beck and Katafygiotis [9] are the pioneering efforts in probabilistic FE odel updating using the Bayesian inference schee. One iportant advantage of Bayesian odel updating ethods to deterinistic ethods is their application in locally identifiable and unidentifiable cases of inverse probles [7]. Another iportant advantage of these ethods is to estiate the level of confidence on the odel updating results. In continuous SHM, continual accuulation of data increases the level of confidence/accuracy in odel updating and therefore daage identification results by itigating the effects of easureent noises and estiation errors as the aount of easured data increases [8]. One of the challenges in the application of Bayesian odel updating ethods is the case of ultidiensional coplex systes with several updating paraeters. These cases involve solving ulti-diensional integrations to calculate arginal posterior probability distributions of odel paraeters or structural response quantities of interest. One coon approach to tackle this proble is generating Markov Chain saples fro the posterior joint probability distribution of the odel paraeters. This nuerical technique has been successfully applied on several nuerical and sall-scale laboratory applications [9-]. Although the deterinistic FE odel updating ethods have been used for daage identification of several real-world, large-scale structures [3-3] including the Dowling Hall footbridge [3], there are few applications of Bayesian FE odel updating ethods to full-scale coplex structures [3-33]. Specifically, application of Bayesian FE odel updating to largescale operational civil structures is very rare due to the existing challenges when dealing with real data and large odeling errors. Most of the probabilistic studies are applied to nuerically siulated data, and in the absence of large odeling errors that can significantly affect the odel updating results. In this paper, the authors investigate the challenges of ipleenting a Bayesian FE odel updating fraework for daage identification of a full-scale structure, the Dowling Hall footbridge. The available vibration data are collected during the operational condition of the structure and therefore contain realistic sources of uncertainty/variability. The ain sources of uncertainties include the variability in the structural ass, stiffness and boundary conditions due to changing pedestrian traffic and environental conditions.

3 Daage on the footbridge is siulated by addition of.9 etric tons of concrete blocks on a sall segent of the bridge deck. The effect of added ass will be siilar to a loss of stiffness (coonly used as daage indication) at the loaded segent of the bridge. The extracted odal paraeters fro abient acceleration tie histories of the daaged structure (loaded structure) are used to find the location and extent of the daage (added ass) through a Bayesian FE odel updating schee. Quantifying the value of added easured data has been recognized as one of the grand challenges in the SHM counity. The value of using continuously easured data on the accuracy and resolution of odel updating results has been quantified in this study. Finally, the results of probabilistic daage identifications are copared to their counterpart deterinistic daage identification. Coparison of the probabilistic and deterinistic FE odel updating results provides insight into the benefits and liitations of the used probabilistic ethod over the deterinistic FE odel updating for structural identification. The footbridge deck is divided into five segents and the added ass on each segent is considered as a odel paraeter to be calibrated in the probabilistic odel updating process. An adaptive Metropolis-Hastings algorith [34-39] is used to saple the posterior probability distributions of the updating paraeters given the easured data and the odel class. Effect of the aount of data used in the updating process is investigated on the accuracy of probabilistic daage identification results. This is done by coparing nine different cases of daage identification results obtained based on,, 6,, 4, 36, 48, 6 and 7 sets of identified odal paraeters. This paper is organized in the following order. In Section, the Dowling Hall Footbridge and its continuous onitoring syste are introduced. Section 3 explains how daage is siulated on the footbridge through addition of ass on a segent of the bridge deck. Brief reviews of Bayesian FE odel updating and the sapling technique used in this study are provided in Section 4 while the odel updating results are reported in Section 5. Finally, the concluding rearks are presented in Section 6.. DOWLING HALL FOOTBRIDGE AND ITS CONTINUOUS MONITORING SYSTEM The Dowling Hall Footbridge is located at the Medford, Massachusetts capus of Tufts University. Figure shows the south view of the footbridge. The bridge is 3.9 wide and consists of two spans. It connects the ain capus on its western end to the Dowling Hall on its eastern end. The footbridge is coposed of a reinforced concrete deck and a steel frae. More inforation about the structural details of the Dowling Hall Footbridge can be found in [4]. A continuous onitoring syste was installed on the footbridge in Noveber 9 and has been providing continuous easureents since January. The onitoring syste consists of eight acceleroeters and a data acquisition device that is connected to the Tufts wireless network. The onitoring syste continuously saples the acceleration channels at a sapling rate of,48 Hz. A five-inute data saple is recorded at the top of every hour or when the root-ean square (RMS) value of an acceleration easureent exceeds.3 g. Locations of acceleroeters on the footbridge are shown in Figure. Details about design and deployent of this continuous onitoring syste can be found in [4]. The odal paraeters of the first six (ost excited) vibration odes identified fro a preliinary test data in April 9 are shown in 3

4 Figure 3. In this figure, the ode shape plots are interpolated between the locations of acceleroeters using a cubic spline. The data-driven stochastic subspace identification ethod (SSI-Data) [4-43] is used for autoatic odal identification of the footbridge. 3. SIMULATION OF STRUCTURAL DAMAGE ON THE FOOTBRIDGE To siulate the effects of daage, a sall segent of the footbridge deck was loaded with.9 etric tons (,9 kg) of concrete blocks for 7 hours. The length of the loaded segent is 4.9 eters. The added ass will cause the sae reduction in the natural frequency of ode one as a 35% loss of stiffness in the sae segent of the bridge. Figure 4 shows the blocks on the bridge deck and safety easures that were considered for this test. 7 sets of abient vibration easureents were collected once every hour and their corresponding odal paraeters representing odel paraeters of the daaged structure were identified through an autoated operational odal analysis fraework. Figure 5 shows the effects of added ass on the identified natural frequencies of odes to 6. The figure also shows the identified natural frequencies 4 hours before and 4 hours after the loading. The grey dots correspond to the undaaged condition while the black dots refer to the daaged (loaded) condition of the bridge. Significant drops of natural frequencies are observed for ode,, 3, and 4 while the natural frequencies of odes 5 and 6 reained alost unchanged. It is worth noting that the drops in the identified natural frequencies can be used for the detection of daage on the structure, however, they do not provide the location and extent of daage. Inforation about the identification success rate and statistics of odal paraeters identified during the loading period are provided in Table. The ost reliably identified odes are odes 3 and 4, while odes 5 and 6 have the largest estiation uncertainty. 4. BAYESIAN FE MODEL UPDATING AND MARKOV CHAIN SAMPLING The first step in the FE odel updating process is to calibrate an initial FE odel of a structure based on its design inforation to a reference FE odel corresponding to as built condition in the undaaged/baseline state of the structure. The initial FE odel of the Dowling Hall Footbridge was created using FEDEASLab [44], a MATLAB-based structural analysis software, based on design drawings and visual inspections of the footbridge. The odal paraeters extracted fro the data recorded at 9 a of July 8, (one day before loading the footbridge) are selected as the reference odal paraeters. The initial FE odel is calibrated so that its odal paraeters best atch to reference odal paraeters using a deterinistic sensitivity-based FE odel updating [3]. The deterinistic FE odel updating is perfored by tuning the updating paraeters of the initial odel to iniize an objective function which represent the isfit between odel-predicted and experientally-identified odal paraeters. The calibrated odel, referred to as the reference FE odel, is assued to represent the true undaaged condition of the footbridge. In the Bayesian FE odel updating perfored in this study, the footbridge deck in the reference FE odel is divided to five segents and the added ass of each segent is considered as an updating paraeter. These five segents are shown in Figure 6. The considered segents in this study are defined based on the location of the sensors, and they are liited to five to assure having a globally identifiable proble. The instruentation of the eastern side of the bridge, nearest to the Dowling Hall, was not possible because of the height above the ground. Low nuber of sensors on the eastern side of the bridge has resulted in selecting a large segent on this side of the bridge, i.e., segent 5. Using saller segents (i.e., 4

5 a larger nuber of updating paraeters) on the eastern side of the bridge can therefore result in locally or globally unidentifiable proble unless ore inforative data (e.g., ore sensors and larger nuber of odes) are used in the updating process. In addition, increasing the nuber of paraeters will deand significantly higher coputational efforts. Alternatively, a different nuber of segents and/or different segent lengths could be selected as updating paraeters which would yield different updating results. FE odel updating results are sensitive to the selection of updating paraeters and this sensitivity can be viewed as one of the ain liitations/shortcoings of this ethod for daage identification. Potentially, ultiple odel updating cases based on different sets of updating paraeters (odel classes) can be perfored and the optial set of updating paraeters can be selected through a Bayesian odel class selection technique [3]. It is also worth noting that in this study the added ass is uniforly distributed within one of the five updating segents which in turn reduces the odeling errors for daage identification. By liiting the aount of uncertainty in the distribution of daage, the perforance of FE odel updating with increasing the aount of easured data can be studied in the presence of odeling errors and variability in the identified odal paraeters. It would also allow coparing the obtained results with the exact daage scenario which would not be available if the added ass was not uniforly distributed along the coplete length of the substructure. In that case, the estiated daage easure will be representing the equivalent unifor change (or any predefined shape when using shape/daage functions [45]) and should be considered as another source of uncertainty (e.g., in odeling error). In real-world applications of odel updating for daage identification, daage is often seared along parts of different updating substructures, which causes increased error in daage identification results. Identification of seared daage along different updating segents of a structure is out of scope of this study. 4.. Bayesian Forulation This section presents a suary of the Bayesian FE odel updating forulation used in this study. More detailed forulation of the Bayesian odel updating process can be found in seinal publications on this topic [7-]. According to the Bayes theore, conditional posterior probability distributions of updating paraeters θ given the easured data d (identified odal paraeters in this study) and the odel class M can be obtained by ultiplying the likelihood function p( d θ, M), the conditional prior probability distribution of odel paraeters p( θ M), and a constant c. p( θ d, M) = cp( d θ, M) p( θ M) () The likelihood function is the probability of easured data given the updating odel paraeters and the noralization constant c is to ensure that the posterior probability density function (PDF) integrates to one. Only one odel class is considered in this study, thus conditioning on the odel class M will be dropped hereafter. Data vector d contains a set of odal paraeters identified fro one set of abient easureents. Modal paraeters are assued to be independently distributed fro ode to ode and fro natural frequencies to ode shapes. In other words, it is assued that knowing the values of any observed odal paraeters do not provide any inforation regarding the probability of observing other odal paraeters. The likelihood function is defined as: 5

6 N p( d θ) = p( λ θ) p( Φ θ) () = where N is the nuber of identified odes, ( ) λ = π f is the identified eigen-frequency and Φ is the identified ode shape of ode. In the case of sall odeling errors, it is reasonable to assue the estiation uncertainties of odal paraeters are independent. However, when considerable odeling errors exist between the reference FE odel and the real syste, the assuption on independency of identified odal paraeters will not be realistic. The eigenfrequency and ode shape errors are defined as: e ( ) λ = λ λ θ (3) Φ ΓΦ( θ) eφ = a (4) Φ ΓΦ ( θ) The atrix Γ picks the observed degrees of freedo fro odel-calculated ode shape Φ( θ ). The scaling factor a is defined in Equation (5).. ( ) a = Φ ΓΦ θ (5) Φ ΓΦ θ ( ) Assuing the errors are zero-ean Gaussian distributed rando variables and the ode shape covariance atrix is a diagonal atrix with all diagonal eleents equal to σ Φ, probability distributions of the identified eigen-frequency and ode shape at a given θ becoe: ( λ ( )) λ θ p( λ θ) exp σ λ (6) T ( ) ( ) p( ) exp Φ ΓΦ θ Φ ΓΦ θ Φ θ a a σ ( ) ( ) Φ Φ ΓΦ θ ΓΦ θ Φ Inserting Equations (6) and (7) into () yields the following likelihood function: p( d θ) exp( J( θd, )) (8) N ( ) λ λ T N = σλ = σ Φ Φ Φ ( θ) Φ ΓΦ ( θ) Φ ΓΦ ( θ) J( θd, ) = + a a ΓΦ ( θ) ΓΦ ( θ) In the case of having independent sets of easured odal paraeters, the probability p( d : d θ ) can be stated as: (7) (9) 6

7 p( d : d θ) = p( d d : d, θ) p( d : d θ) =... = p( d θ) Nt Nt Nt Nt n n= Nt = cˆ exp J( θd, n) n= where it is assued that the uncertainty in the n th data set is not affected by previous data sets, i.e., p( dn d: dn, θ) = p( dn θ ). The posterior probability distribution of odel paraeters in Equation () is a joint PDF, and therefore a ulti-diension integration is required to obtain the arginal probability distribution of each odel paraeter: p( θ d) = pθ (, θ d) dθ i i -i -i Θ-i where subscript -i refers to all the updating paraeters except the i th paraeter. These probability distributions often cannot be calculated analytically and are usually estiated nuerically. Markov Chain Monte Carlo (MCMC) ethods are usually ipleented to saple such coplex ulti-diensional probability distributions. MCMC is a Monte Carlo sapling technique where saples are correlated through a Markov chain. In Monte Carlo ethod, saples are drawn independently fro a target probability distribution. In soe applications, generating independent and identically distributed saples fro the target distribution is not feasible or coputationally expensive; therefore, dependent saples can be used instead. One class of generating such dependency in a sequence of saples is Markov Chain. In Markov Chain Monte Carlo, the next saple is selected fro a distribution that depends only on the current saple. 4.. Sapling the Posterior Probability Distributions of Model Paraeters Aong the MCMC ethods, the Metropolis-Hastings (MH) algorith is the ost coon ethod used in the application of FE odel updating [9-]. This algorith is based on generating saples fro a prescribed proposal PDF and accepting the saples with a probability of ove. One intuitive choice for the proposal probability distribution is a localized, syetric PDF at the current saple. In this case, the algorith can be viewed as a rando walk process that continues until the high probability region of the target PDF is sufficiently explored. However, direct application of the standard MH algorith is not suited for cases where high probability regions of odel paraeters are concentrated in a sall volue of the paraeter space or in the cases with ulti-odal target PDFs. To be able to address these difficulties, different adaptive Metropolis-Hastings (AMH) algoriths have been proposed. The adaption can be perfored on either the proposal probability distribution or the target probability distribution. The algoriths with adaption on the target PDF, such as the adaptive algorith used in [9] or the TMCMC algorith [], incorporate a nuber of interediate target probability distributions that converge to the posterior PDF of Equation (). The advantage of using these ethods is their ability to saple the whole space and explore all the peaks (high probability regions). However, the ain disadvantage of these ethods is the high coputational cost required in the case of coplex structural odels where the nuber of updating paraeters increases. Another approach is to adapt the proposal PDF [37-38]. The advantage of this ethod is that it is fast for identifiable cases where the posterior distributions of updating paraeters Nt () () 7

8 have unique clear peaks. In these cases, saples quickly converge to the highest probability region even when the region is sharply peaked. In this study, the adaption is perfored on the proposal PDF instead of the target PDF. The N θ, β Σ, for the algorith is briefly introduced below assuing a Gaussian distribution ( ) proposal probability distribution function. Initialize θ, μ, and Σ, which are the vectors of initial paraeter values, the ean vector, and the covariance atrix of the initial proposal probability distribution, respectively. At iteration j+ given θj, μ j, Σ j, and β j : Generate saple j+ ( j, j ) α + ζ fro N ( j, β j j) θ ζ. If ζ j+ is not accepted then set θ = θ j+ j. where α ( θj, ζ j+ ) ( ) ( ) θ Σ and set θ = ζ with probability of ove 8 j+ j+ j j+ j j j j ( ) ( ) p ζ j+ d N( θj ζ j+, β jσ j) p ζ j+ d = = p θ d N( ζ θ, β Σ ) p θ d Update the scale factor β j +, ean μ j+, and covariance atrix Σ j+ of the proposal PDF as: The step size sequence ( ) ( ) j+ j j+ ( θj ζ j+ ) = + γ ( ) γ ( )( ) * log β = log β + γ α, α μ μ θ μ j+ j j+ j+ j Σ = Σ + θ μ θ μ Σ j+ j j+ j+ j j+ j j soe β >, and here it is taken as γ j needs to satisfy two conditions of T + β γ j = and γ j < j= j=. γ j = j which satisfies both conditions. The paraeter is the desired acceptance ratio and is taken as 44% [38]. The optial acceptance rate for one diensional Gaussian target probability densities is 44% while the optial rate will becoe 3% when the diension of the target distribution goes to infinity [39]. 5. FE MODEL UPDATING RESULTS The first case of Bayesian FE odel updating is based on the average of identified odal paraeters over the 7 sets of data and the results are presented in Section 5.. In section 5., nine cases of odel updating are perfored using different subsets (,, 6,, 4, 36, 48, 6, and 7 sets) of available identified odal paraeters. In all the FE Bayesian odel updating cases,, Markov Chain saples are generated. The nuber of burn-in saples is,. The prior distributions of segents added asses are considered as independent unifor distributions with the lower bound of and upper bound of 6.8 tons. Values of the standard deviations for eigen-frequencies in Equation (9) are considered as: σ = w COV λ (4) λ λ () (3) for * α

9 where COV λ is taken as the average of calculated eigen-frequency coefficients-of-variation (COVs) of all the six odes (the COVs of natural frequencies are reported in Table ) and w is a weight associated to each ode, taken as for odes, 5, and 6 to account for their higher estiation uncertainties and one for the other odes. This factor is selected based on prior knowledge of syste identification studies of this footbridge [5, 3, 4]. Standard deviations of ode shapes are coputed fro: σ Φ = Nw COV (5) s λ where N s is the nuber of acceleroeters, or nuber of identified ode shape coponents, i.e., N s = Bayesian Updating Results Based on Average Modal Paraeters The posterior probability distributions of the added ass at the five considered segents of the footbridge are sapled using the aforeentioned sapling process. Figure 7 shows the saples versus their posterior probability densities. The burn-in saples are excluded fro the graphs. The axiu a-posteriori (MAP) estiates of the updating paraeters are shown by black dots and the exact values of the added ass are shown by stars. The convergence of the sapling process is confired by the convergence of their eans, standard deviations, and the correlation between the saples. Figure 8a shows the cuulative ean values of saples generated at updating paraeters and (i.e., added asses on segents and ). The cuulative eans vary significantly when the total nuber of saples is sall but its value becoes alost constant as the nuber of saples increases beyond 6,. Figure 8b shows the cuulative standard deviations of the saples and Figure 8c plots the correlation coefficients between the saples. For both updating paraeters, the correlations becoe negligible beyond 6, saples. Table provides the statistical inforation of the ean, standard deviation, and MAP of the five odel paraeters based on the MCMC saples as well as the MAP estiates when using Gaussian kernel functions to fit the posterior probability distributions of the updating paraeters. The updated odel paraeters obtained fro a deterinistic FE odel updating are also listed in the table. Fro this table and Figure 7 the following observations can be ade. For segents, 3, 4 and 5, the axiu a-posteriori of added ass values are close to zero, which is expected as no ass is loaded on these segents. For segent, the posterior probability density of added ass is the largest in the range of. to.5 tons, with the MAP of.9 tons (which is equal to the exact value). The posterior probability distributions of the added ass on segents, 3, 4, and 5 are onesided because of the lower bound constraint of zero for these updating paraeters ipleented in the prior PDFs. The posterior PDF of segent is two-sided but not syetric; it is skewed to the left. This ay be due to the fact that larger values of θ have to be copensated by negative values of other updating factors which are not allowed. This has resulted in large bias in the saples ean for each of the odel paraeters. The estiates of paraeters θ to θ 4 have larger variances than that of θ 5. This is due to the higher sensitivity of updating paraeter θ 5 to the considered odal data, i.e., larger observability of this paraeter in the view of data. Alternatively, this observation could be 9

10 reached intuitively as this is the largest segent of the bridge deck and therefore has the largest effect on the odal paraeters of the footbridge. The deterinistic estiates of odel paraeters are in relatively good agreeent with the MAP estiates. Theoretically, the deterinistic and MAP estiates should be identical when using copatible objective/likelihood functions. However, in this application due to liited nuber of data sets used (i.e., one) and saller nuber of saples, there is soe discrepancy between these two estiates. The MCMC saples are used to construct the posterior PDF of each odel paraeter instead of calculating the integral of Equation () analytically. Gaussian kernel functions are used to estiate the arginal posterior PDF of each odel paraeter. Figure 9 shows the noralized envelope of sapled arginal posterior PDFs (envelope of Figure 7) and the kernel arginal posterior PDFs of updating paraeters. The envelope of saples at each point corresponds to the axiu of the joint posterior PDF for the specific value of the considered paraeter. The kernel PDF estiate is based on suation of noralized Gaussian kernel functions for each of the saple bins and a weight associated with each saple. The weights of the saples are taken as the posterior probability of the and the bin widths in this section are considered as.8 tons for θ and θ 4, and.5 tons for θ, θ 3, and θ 5. The bin width for each updating paraeter is selected based on the observed range and standard deviation of saples; therefore, different bin widths are used for different paraeters. It is worth noting that the kernel probability distribution estiates fail to accurately represent the MAP and standard variation of saples. As it will be shown and discussed in the next section, the kernel PDF estiates provide uch better representation of the saple probability distributions as the nuber of data sets used in the likelihood function increases. 5.. Bayesian Updating Results Based on Subsets of Identified Modal Paraeters In this section, effects of the nuber of data sets used in the likelihood function on the accuracy of updating results are investigated. Nine cases of probabilistic FE odel updating are perfored with =,, 6,, 4, 36, 48, 6, and 7 data sets (see Equation ()), corresponding to the first sets of identified odal paraeters. As expressed in Equation (), using sets of data is equivalent to the ultiplication of likelihood functions, each including one set of identified odal paraeters. Figure shows the envelope of generated Markov Chain saples with the MAP of saples in each bin shown by black dots, the kernel arginal posterior PDFs, and the approxiated Gaussian posterior PDFs for the updating odel paraeter θ and nine cases of odel updating ( =,,, 7). The light grey lines in this figure are approxiated as a Gaussian distribution with ean and standard deviation of the Markov Chain saples. Alternatively, the posterior probability distribution of the updating odel paraeters can be shown by the cuulative distribution function (CDF) of saples. The probability that added ass at the i th segent is lower than L tons is estiated using the N c generated Markov Chain saples: P L d P L d H L (6) N c AM i ( ) i i j Nc j= { θ } ( θ, ) where H(z) is a unit step function, defined as H(z) = if z > and H(z) = if z. Figure shows the CDFs of the added ass at segents, 3, 4, and 5 for the nine probabilistic FE odel

11 updating cases. The MAP, ean, and standard deviation estiates of the posterior probabilities of all five paraeters are reported in Table 3. The standard deviation and the bias between the MAP and saple eans as function of data points ( ) are plotted in Figures and 3, respectively, for the five paraeters. Fro Figures -3 and Table 3, the following observations can be ade. The estiation uncertainties (i.e., standard deviations) of the updating paraeters significantly decrease as ore data is used in the Bayesian odel updating process. However, this reduction becoes less significant as the nuber of data sets exceeds 36. Therefore, it is expected that additional data (ore than 7 sets) would not drastically iprove the estiation accuracy of updating paraeters. Such inforation can be used to quantify the value of additional data for paraeter estiation. The three estiates of the posterior probability distribution of each odel paraeter, naely the saple envelope, the kernel arginal PDF, and the approxiated Gaussian PDF are very different for the case =, but these estiates get closer to each other as ore data sets are used. The three posterior PDF estiates are in good agreeent for the case of = 7 with MAP values of.48,.49, and.49 tons. It should be noted that although the saple envelope and kernel probability distribution estiate represent different inforation about probability distribution of an updating paraeter, these two types of inforation becoe siilar when the updating paraeters are independent. Therefore, the closer the kernel PDFs represent the MAP and standard deviation of the saples, the saller is the correlation between updating paraeters. With increasing the nuber of data sets, the posterior PDFs becoe ore syetrical and therefore the discrepancies between the saple eans and MAP estiates are reduced. The standard deviation of odel paraeter θ 4 is uch larger than the standard deviations of paraeters θ, θ 3, and θ 5, especially when. This can be due to the fact that this segent is located on the iddle support and has saller vibration responses and therefore, the identified odal paraeters are less sensitive to this updating paraeter. Although the identification ethod can successfully predict the location and extent of the added ass on the footbridge, the MAP estiate of added ass on segent has soe bias fro the actual ass of.9 tons. This bias in the MAP estiation (.9 tons for the case of = 7) is ainly due to the odeling error in the reference FE odel and the variability in the identified odal paraeters caused by changing environental conditions and estiation errors. Figure 4 shows the posterior correlation coefficients between updating paraeters for the nine different cases of odel updating. The correlation coefficients are high when few data sets are used, but they reduce as increases. In other words, the updating paraeters are sapled ore independently when ore data sets are used (axiu correlation coefficients is calculated as.7 when 36). To quantify the effects of easured data on odel updating results, an eigen-analysis is perfored on the covariance atrices of prior and posterior joint PDFs [33, 46-47]: Σ X = ΛΣ X (7) po pr

12 where the eigenvectors X are orthogonal directions in the paraeter space ranked based on the corresponding eigenvalues. The eigenvalues give a easure of reduction in variances fro prior covariance atrix Σ pr to posterior covariance atrix Σ po. Figure 5 shows the eigenvalues coputed fro Equation (7) for the nine odel updating cases. The changes in the entropy of updating paraeters fro prior to posterior quantify the value of easured data on the accuracy of updated paraeters and can be estiated as: Nθ h= h h.5 log λ (8) pr po k k = where h refers to inforation entropy, and λk is the k th eigenvalue coputed fro Equation (7). Each ter of.5log λ k represents the relative contribution of the corresponding eigenvalue to the total entropy reduction fro the prior to the posterior probability distribution of updating paraeters. The last eigenvalue has little contribution to the total entropy reduction when is sall, but its contributions becoes ore significant as increases. For exaple, the fifth eigenvalue accounts for only 9.7% of the total entropy reduction when =, but contribution of λ 5 becoes 7.7% for =7. Inversely, contribution of λ is 9.3% when = and is reduced to.% when =7. The inforation entropy reduction is ore sensitive to the nuber of data sets when is sall but this reduction becoes less significant for cases of using ore than 36 data sets. This observation is consistent with the reduction in standard deviations and bias of odel paraeter estiates in Figures and 3. It is observed that using different subset of data with the sae nuber of data sets will provide siilar posterior probability distributions of odel paraeters but with slightly different MAP estiates. Figure 6 shows the posterior PDFs of the added ass on segent using three different subsets of identified odal paraeters with = 4. The first subset corresponds to the data collected during the first 4 hours of loading, the second set contains the second 4 hours of data, and the third set is the last 4 hours of the easured data during the loading period. Although the three PDFs have alost the sae standard deviations (.44,.44,.43 tons), their MAP estiates are slightly different (.5,.47, and.5 tons). In practice, structural daage is quantified by coparing the updating odel paraeters in the daaged and undaaged states of a structure. Therefore, it is iportant to recognize that the posterior PDFs or MAP estiates of the updating odel paraeters are conditional on the easured data and therefore, depend on the accuracy/uncertainty of easureents in both the daaged and undaaged conditions. It should be noted that the changes in environental conditions such as air teperature ay have significant effects on the identified odal paraeters, especially when the abient teperature drops below the freezing point [3]. These effects can consequently affect the odel updating results and should be taken into account [3] when large variations in environental conditions are observed. In the current application, the environental conditions reained relatively consistent during the week of testing with teperatures being in the range of 7 to 33 degrees Celsius. The obtained bias in odel updating results could be reduced by including the effects of odeling errors in the Bayesian forulation. For exaple, Sohn and Law [6] assued that the ean values of the natural frequency errors and ode shape errors in Equations (3) and (4) can be approxiated by the corresponding ean values in the undaaged state. Beck and coworkers [48] included the effects of odeling error in their forulation by introducing syste

13 ode shapes and syste eigen-frequencies. Syste ode shapes are also used in [8] and []. In this study, the odel updating results are found to be sensitive to the considered standard deviations of the identified natural frequencies and ode shapes in Equations (4) and (5), i.e., the posterior probability distributions obtained are conditional on the σ λ and σ Φ values. The odel updating results are sensitive to both the relative standard deviations aong different odes and the relative standard deviations between the natural frequency and ode shape of each ode. Soe studies have been perfored to itigate the effects of this uncertainty [49]. Finally, to show the coputational efficiency of the adaptive sapling algorith used in this study, the generated sequence of saples for paraeters and and the case of = 7 are plotted in Figure 7. It can be seen that after only 3, iterations the saples reach the high probability region of the paraeter space. The nuber of saples required to reach convergence was found to be uch larger when using the standard MH algorith. The acceptance ratios of all runs are close to 44%, which is the desired acceptance ratio used in Equation (3) Deterinistic FE Model Updating Results Deterinistic FE odel updating is perfored using the easured daaged data through a global optiization approach, naely the ultistart function fro the MATLAB global optiization toolbox [5]. This optiization algorith is based on several Gauss-Newton sensitivity-based optiizations with different starting points to ensure global optiality. Equation (9) is considered as the deterinistic objective function to be iniized. The axiu nuber of iterations for each optiization is liited to 8 and axiu nuber of initial points is assigned as. Figure 8 shows the scatter of the updating paraeters for all 7 deterinistic updating cases. It is observed that, except for a few outliers, the added ass on segents, 3, and 4 are accurately estiated as zero with no variability. In this case, the outliers are defined as identification cases with nonzero ass on either segent, 3, or 4. However, the added ass on segents and 5 are estiated with larger variability. The ain sources of estiation errors in updating paraeters are incopleteness of identified odal paraeters fro the corresponding easured data. The odal paraeters in the outlier cases are incoplete (at least issing a ode) and are identified with larger estiation errors, which yield to inaccurate odel updating results. It is also observed that when the first ode is issing (i.e., could not be identified) in a data set, relatively larger added ass is predicted at segent. These large estiation errors are avoided in the Bayesian approach when ultiple data sets are used in one updating process. 6. CONCLUSIONS A Bayesian FE odel updating approach is ipleented for identification of physically siulated daage on the Dowling Hall Footbridge. The extent and location of the daage (added ass) was accurately estiated based on the continuously easured vibration data. The probability distributions of the updating paraeters do not only provide an estiate for the location and extent of daage but also a easure of confidence/uncertainty of the daage identification results. The MAP estiates of updating odel paraeters atch the exact values and are in a good agreeent with the optiu values fro the deterinistic FE odel updating. Effects of the nuber of data sets used in the identification process (i.e., value of added data) are 3

14 investigated by using different subsets (,, 6,, 4, 36, 48, 6, and 7) of available data. Estiation uncertainty of the updating odel paraeters are significantly reduced by adding ore data sets to the likelihood function, which iplies ore accurate odel updating results. However, this reduction becoes less significant as the nuber of data sets exceeds 36. Therefore, it is expected that additional data (ore than 36 sets) would not drastically iprove the estiation accuracy of updating paraeters. Such inforation can be used to quantify the value of additional data for paraeter estiation. Adding ore data sets also affects the shape of the posterior PDFs of updating paraeters resulting in saller bias between the saple eans and the MAP estiates of odel paraeters. It is also worth noting that in the application of deterinistic FE odel updating, addition of ore data sets will not necessarily iprove the odel updating results. Thus, probabilistic FE odel updating approaches based on ultiple sets of easureents are strongly recoended for structural identification purposes to yield ore accurate prediction and detection of potential structural daage. Although the ipleented FE odel updating ethod has successfully estiated the location and extend of daage in this study, but in general the success of this ethod depends on the accuracy of the initial FE odel, the selected updating paraeters, and the considered residuals and their weights in the objective function. Sensitivity of daage identification results to these factors can be viewed as one of the ain liitations of this ethod for ipleentation by the practicing engineers without experience in odel updating and inverse probles. For a robust identification, however, different cobination of these factors (i.e., initial odels, updating paraeters, and objective functions) can be considered as different odel classes, and a Bayesian odel class selection/averaging technique can be used to select the optial set of these factors. ACKNOWLEDGEMENT The authors would like to acknowledge support of this study by the National Science Foundation Grant No. 564 which was awarded under the Broadening Participation Research Initiation Grants in Engineering (BRIGE) progra. The authors also acknowledge Ms. Alyssa Kody for the design and perforance of the bridge load test, Ms. Rachele Pesenti for her help in prograing the real-tie odule of the data acquisition syste, and Mr. Bidiak Aana for his help in setting up the wireless network connection of Dowling Hall footbridge. The opinions, findings, and conclusions expressed in the paper are those of the authors and do not necessarily reflect the views of the individuals and organizations involved in this project. REFERENCES [] National Transportation Safety Board (NTSB). Collapse of l-35 Highway Bridge 8, Washington, Rep. NO. PB [] Aerican Society of Civil Engineers (ASCE). Report Card for Aerica's Infrastructure 9. Available fro: [3] Rytter A. Vibration based inspection of civil engineering structures. Ph.D. dissertation, Departent of Building and Technology and Structural Engineering of Aalborg University 993, Denark. 4

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17 [34] Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. Equations of state calculations by fast coputing achines, Journal of Cheical Physics 953; : [35] Hastings WK. Monte Carlo sapling ethod using Markov Chains and their applications, Bioetrika 97; 57: [36] Chib S, and Greenberg E. Understanding the Metropolis-Hasting algorith, The Aerican Statistician 995; 49(4): [37] Haario H, Saksan E, Tainen J. An adaptive Metropolis algorith, Bernouli ; 7: 3-4. [38] Andrieu C, and Thos J. A tutorial on adaptive MCMC, Statistics and Coputing 8; 8: [39] Roberts GO, and Rosenthal JS. Optial scaling for various Metropolis-Hastings algoriths, Statistical Science ; 6(4): [4] Bowan J. Vibration Testing and Modal Identification of the Dowling Hall Footbridge at Tufts University, Master s thesis, Departent of Civil and Environental Engineering of Tufts University 3, Medford, Massachusetts. [4] Moser P, and Moaveni B. Design and deployent of a continuous onitoring syste for the Dowling Hall Footbridge, Experiental Techniques ; 37(): 5-6. [4] Van Overschee P, and De Moor B. Subspace identification for linear systes, Massachusetts, 996, Kluer Acadeic Publishers. [43] Peeters B, and De Roeck G. Reference-based stochastic subspace identification of output only odal analysis, Mechanical Systes and Signal Processing 999; 3 (6): [44] Filippou FC, and Constantinides M. FEDEASLab getting started guide and siulation exaples 4, Technical Report NEESgrid-4-. Available fro: [45] Teughels A, De Roeck G. Daage detection and paraeter identification by finite eleent odel updating, Archives of Coputational Methods in Engineering 5; (): [46] Papadiittriou C, Beck JL, Au S. Entropy-based optial sensor location for structural odel updating, Journal of Vibration and Control ; 6(5): [47] Papadiittriou C. Optial sensor placeent for paraetric identification of structural systes, Journal of Sound and Vibration 4; 78: [48] Yuen KV, Beck JL, Katafygiotis LS. Efficient odel updating and health onitoring ethodology using incoplete odal data without ode atching, Structural Control and Health Monitoring 6; 3: 9-7. [49] Goller B, Beck JL, Schueller GI. Evidence-based identification of weighting factors in Bayesian odel updating using odal data, Journal of Engineering Mechanics ; 38: [5] Mathwork Inc. Global Optiization Toolbox: User s Guide, Natick,. MA. 7

18 Tables: Table. Statistics of identified odal paraeters during loading period Mode N a f 3 4 µ σ f f COV Total nuber of available odal paraeters. Mean of available natural frequencies [Hz]. 3. Standard deviation of available natural frequencies [Hz]. 4. Coefficient of variations of available natural frequencies [%]. i 5. Standard deviation of available unit-length noralized ode shapes; defined for ode as: σ = Φ Φ Φ. σ Φ Na N a i= Table. Statistical inforation of odel paraeters when using average of odal paraeters in the likelihood function θ θ θ 3 θ 4 θ 5 MAP [tons] Mean of Saples [tons] STD of Saples [tons] MAP of kernel arginal PDF [tons] Mean of kernel arginal PDF [tons] STD of kernel arginal PDF [tons] Optia fro deterinistic FE odel updating [tons]

19 Table 3. Statistics (axiu a-posteriori [tons], ean [tons], and standard deviation [tons]) for five updating paraeters and nine considered cases of odel updating No. of data sets θ θ θ 3 θ 4 θ 5 MAP ean STD MAP ean STD MAP ean STD MAP ean STD MAP ean STD

20 Figure. South view of the Dowling Hall Footbridge.

21 North Dowling Hall Upper Capus Figure. Layout of acceleroeters on the bridge.

22 f = 4.63 Hz, ξ =. % f = 6.7 Hz, ξ =.6 % 3 f 3 = 7.7 Hz, ξ 3 =.7 % 4 3 f 4 = 8.9 Hz, ξ 4 =.3 % 4 3 f = 3.3 Hz, ξ =.8 % f = 3.56 Hz, ξ =. % Figure 3. Identified odal paraeters fro preliinary test data.

23 Figure 4. Concrete blocks loaded on footbridge s deck for three days.