Proceedings of the 9th International Conference on Structural Dynamics, EURODYN... Porto, Portugal, 30 June - 2 July 2014
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4 Bayesian model updating based on modal flexibility for structural health monitoring Z. Feng 1, L.S. Katafygiotis 2 Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 2 Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China Email: [email protected], [email protected] 1 ABSTRACT: A Bayesian probabilistic model updating method based on modal flexibility for structural health monitoring is presented. Firstly, the flexibility matrices are constructed by using a sequence of identified modal data sets including modal frequencies and mode shapes. Then flexibility vectors are obtained by using vectorization of the lower triangular portion of the flexibility matrices and their covariance matrix is also calculated. The obtained flexibility vector data sets and their covariance matrix are incorporated into a formula of Bayesian updating and the most probable values of model stiffness parameters are obtained by maximizing the posterior PDF of the model parameters. The associated uncertainties are also quantified by calculating the covariance matrix of model parameters at the most probable values. By comparing the updated PDFs of stiffness parameters of the structure before and after possible damage, the probability of damage in each sub-structure can be estimated. The proposed method is illustrated with a numerical example. KEY WORDS: Bayesian model updating; modal flexibility; Structural health monitoring. 1 INTRODUCTION There has been a significant amount of research effort devoted to vibration-based methods for structural health monitoring during recent years [1]. Changes in the physical properties of a structure result in changes in the modal properties (frequencies and mode shapes). Many damage detection methods are based on the changes in natural frequencies and mode shapes occurring during damage. Doebling et al. presented a review of the main methods for damage detection based on modal parameters [2]. Methods based on changes in natural frequencies are very attractive since this parameter can be determined by measurement at only one point of the structure [3]. However, changes in natural frequencies cannot provide spatial information about structural changes. Mode shapes which provide spatial information can be used for localizing the damage [4]. Nonetheless, an accurate characterization of these mode shapes requires measurements at several locations and changes in mode shapes due to structural damage are not very significant. Besides modal frequency and mode shape based methods, another class of damage identification methods uses the dynamically measured modal flexibility matrix. Damage identification based on modal flexibility matrix has been recently shown to be promising. Since an inverse relationship exists between the modal flexibility matrix and the square of the modal frequencies, the modal flexibility matrix is not sensitive to high frequency modes. This unique characteristic allows the use of a small number of truncated modes to construct a reasonably accurate representation of the flexibility matrix. Toksoy and Aktan proposed a bridgecondition assessment method which is formulated based on modal flexibility for evaluating the global state of bridge health [5]. Pandey and Biswas employed the changes in modal flexibility matrix to detect damage in structures [6]. More recently, Bernal proposed a flexibility-based damage localization method termed as DLV [7]. Jaishi and Ren proposed a sensitivity-based finite element model updating method using modal flexibility residual [8]. Damage identification techniques using modal data are often based on methods of model updating. A nominal parametric model of the structure is needed and the model parameters are updated by minimization of some objective function which reflects the errors between the measured data and the predictions of the model. The success of the finite element (FE) model updating method depends on the accuracy of the FE model, the quality of the modal data, the choice of the objective function and the capability of the optimization algorithm. Most model updating methods are based on a onestage optimization scheme. When these methods are applied to large-scale structures with many unknowns, ill-conditioning and non-uniqueness in the solution of such inverse problem appear as inevitable difficulties. Furthermore, a large computational effort is usually required. Generally speaking, due to the limited number of sensors and the difficulty of obtaining measurements for rotational DOFs and internal DOFs of the structure, the number of DOFs in the FE model usually exceeds that of the experimental model. Therefore, in order to solve the problem of mismatch between the DOFs of the FE model and those of the experimental model, model reduction or modal expansion is needed. In addition to the aforementioned concerns, accounting for measurement and modeling errors and uncertainties is crucial when applying FE model updating techniques. One possible approach to incorporate uncertainty regarding measurement and modeling errors into the FE model updating process is to adopt a probabilistic scheme based on Bayesian inference [9, 10]. The Bayesian inference approach has gained interest among uncertainty quantification methods in recent years, mostly because of its solid foundation on probability theory and its rigorous treatment of uncertainties. 177 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 In this paper, a Bayesian model updating method based on modal flexibility is proposed. Firstly, flexibility matrices are constructed by using a sequence of identified modal data sets including modal frequencies and mode shapes. Then flexibility vectors are obtained by using vectorization of the lower triangular portion of the flexibility matrices and their covariance matrix is also calculated. The obtained flexibility vector data sets and their covariance matrix are incorporated into a formula of Bayesian updating and the most probable values of model stiffness parameters are obtained by maximizing the posterior PDF of the model parameters. The associated uncertainties are also quantified by calculating the covariance matrix of the model parameters at the most probable values. By comparing the updated PDFs of stiffness parameters of the structure before and after possible damage, the probability of damage in each sub-structure can be estimated. In order to facilitate the optimization problem, a deterministic initial updating stage is added before the Bayesian updating stage with the purpose of identifying an initial estimate of the model stiffness parameters. The initial estimates obtained in the preliminary updating stage can facilitate the optimization problem in the Bayesian updating stage, which makes the algorithm more efficient and robust. Furthermore, the SEREP model reduction technique is adopted to cope with the problem of mismatch between the DOFs of the FE model and those of the experimental model [11]. The proposed procedure is illustrated with a simulated example. 2 MODAL FLEXIBILITY The generalized eigenvalue equation for a linear dynamical system with N degrees of freedom is: K M (1) where M and K are the mass and stiffness matrix; Φ is the eigenvector (mode shape) matrix and Λ is the diagonal eigenvalue matrix with squared modal frequencies . With the modal frequencies and the mass-normalized mode shapes, the stiffness matrix K and flexibility matrix F of a structure can be calculated by [12] N T K M T M M ( i2 i i )M Nm F Ft i 1 N i 1 1 T 2 i i Nm F m Ft m i 1 where i It can be seen from Equation (2) that the modal contribution of one mode to the stiffness matrix increases as the corresponding modal frequency increases. To obtain an accurate estimate of the stiffness matrix, one has to use highfrequency modes. Contrarily, it can be seen from Equation (3) that the modal contribution of one mode to the flexibility matrix is inversely proportional to the square of the corresponding modal frequencies, implying that it can be estimated with sufficient accuracy by using a few lowfrequency modes. Therefore, the Equation (3) can be approximated as 178 iiT (4) 1 2 i im R No1 ( i 1,2,..., Nm ) im (im )T (5) are spatially incomplete mode shapes and No denotes the number of measured DOFs at the sensor locations. It should be noted that im is a vector comprised by the components of the mass-normalized i in Equation (4), corresponding to the measured DOFs. As we known, the flexibility matrix is constructed from modal frequencies and mass-normalized mode shapes, so the key issue for the flexibility matrix construction is massnormalization of the experimentally identified mode shapes. Numerous researchers have developed various methods for mass normalization of the ambient vibration mode shapes. In the present study, the mass-normalized mode shapes are achieved with the aid of the finite element model (FEM). The mass matrix is assumed to be known with reliable confidence. If the measurements are complete, then the massnormalization is trivial. If the measurements are incomplete, then the mass-normalization will be completed with the aid of model reduction techniques. Herein, the SEREP reduction method is adopted for this purpose [11]. The SEREP method is first applied to the FEM mass matrix to obtain a reduced mass matrix Mm with the same dimension as the number of measured DOFs. Then the measured incomplete mode shape vectors are normalized as (2) (3) 2 i where Ft is the truncated flexibility matrix and Nm is the number of selected lower modes. In practice, only modal frequencies and mode shapes of a few lower modes are actually obtained during vibration testing. In addition, usually only spatially incomplete mode shapes, comprised of the mode shape components corresponding to the measured DOFs, which are less than the analytical DOFs, are available. Let Fm denotes the flexibility submatrix with respect to the measured DOFs, Ftm denotes the truncated flexibility submatrix, then (m )T M mm I i 1 F 1T 1 where 3 (6) m [1m , 2m ,..., Nmm ] . MODEL REDUCTION System equivalent reduction and expansion process (SEREP) was proposed by O’Collahan et al. [11]. SEREP is a model reduction or modal expansion technique, in which the reduced system preserves the frequency and mode shapes of the original system for selected modes of interest. This method first partitions the mass and stiffness matrices as follows, M M p mm M sm M ms p K mm ,K M ss K sm K ms K ss (7) Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 The subscripts m and s stand for the master (measured) and slave (unmeasured) DOFs, respectively. Let Φm denotes the retained mode shape matrix, which is a rectangular matrix of size No×Nm, where No is the number of master DOFs and Nm is the number of modes to be retained. In this method, the matrix (Φm)† called the pseudo-inverse of Φm needs to be computed to construct the SEREP transformation matrix. m T m m T 1 M T M T, K T K T T p m T p e T m Substituting Equation (19) into Equation (18), one obtains (12) The subscripts nom and exp stand for the nominal and updated experimental structures, respectively. Let K and F be the perturbation matrices such that the nominal model matrices and the updated experimental model matrices are related as follows: Kexp Knom K (13) Fexp Fnom F (14) C p,n ( Fnom Kn Fnom ) p (15) As the matrix K nom is full rank, then Equation (15) can be (21) and b is a vector of size (NoNo)1 with pth component bp Fp (22) where p is a new index with p = (i,j), that corresponds to the (i,j) entry of the matrix. Because of the symmetry of the modal flexibility matrix, only upper or lower triangular matrix is useful. Thus the sizes of the matrix C and vector b are reduced to [No(No+1)/2]Nθ and [No(No+1)/2]1 respectively. The stiffness factors can be solved as (CT C )1 CT b (23) Using the above approximations, one can obtains initial estimate of the model parameters. 5 BAYESIAN MODEL UPDATING BASED MEASURED MODAL FLEXIBILITY DATA 5.1 ON Formulation Using model reduction as described in the previous section, the number of DOFs for the analytical model can be equal to the number of the measurements. For clarity, the size of modal flexibility matrix for the analytical model is the same as that for the experimental model. Because of the symmetry of the modal flexibility matrix, only the upper or lower triangular matrix is useful. Here we use the lower triangular modal flexibility matrix Fl for model updating. We perform vectorization on Fl as f F11 , F21 ,...FNo 1 ,..., Fii , F(i 1)i ,...FNoi ,..., FNo No Substituting equations (13) and (14) into (12) yields (20) where C is a matrix of size (NoNo)Nθ with (p,n) entry (11) For the nominal and the updated experimental structures, the global stiffness and flexibility matrices will satisfy the following relationship FKnom Fexp K C b (10) INITIAL MODEL UPDATING BASED ON MODAL FLEXIBILITY CHANGE Fnom Knom Fexp Kexp I (19) n 1 (9) Ignoring inherent numerical errors, this SEREP reduction technique usually produces identical eigenvalues to those of the full model and identical expanded eigenvectors to those of the full model. Furthermore, this method can arbitrarily select the modes preserved in the reduced model and the reduced model accuracy does not depend on the selection of master DOFs. rewritten as (18) K n K n The retained mode shape matrix can also be expanded to full mode shape matrix of size N×Nm using 4 F Fnom KFnom The stiffness perturbation matrix K is expressed as The reduced mass matrix and stiffness matrix are then given by m (17) m T ( ) ( ) ( ) , N o N m m † F ( Fnom F )KFnom ( FnomKFnom F KFnom ) N 1 ( ) ( ) ( ) , N o N m m Substituting Equation (14) into (16) yields (8) where the pseudo-inverse can be computed as m T (16) Neglecting the high-order term in Equation (17), one obtains † m T s m m † F Fexp K ( Knom )1 Fexp KFnom T (24) The experimental data D from the structure are assumed to consist of Nt sets of modal flexibility vector data, D { fˆ1, fˆ2 ,..., fˆNt } . fˆt f ( ) et (25) 179 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 where f(θ) is the prediction from the analytical model and et is a prediction error term. The choice for the probability model of the prediction error is based on the maximum entropy principle [13] which yields a multi-dimensional Gaussian distribution with zero mean and covariance matrix , that is et ~ N (0, ) . The covariance matrix is estimated by computing the sample covariance matrix of multiple modal flexibility data sets. Here it is computed based on Nt modal flexibility vector data sets as 1 NT ˆ ( ft f )( fˆt f )T Nt 1 t 1 Assuming independence between these Nt modal flexibility vector data sets, the likelihood function for the whole data sets D can be expressed as 1 Nt p( D , M ) c2 exp [ fˆt f ( )]T 1[ fˆt f ( )] 2 t 1 (28) Assuming a non-informative uniform distribution for the prior PDF p(θ), the posterior PDF p( D, M ) can be written in a similar form as the likelihood function in Equation (28). It should be noted that if the measurements are incomplete, then the measured modal flexibility vector data are also fˆt = fˆt ( ) . Since the model reduction transformation matrix T is dependent on the stiffness matrix K which is a function of model parameters θ, then the reduced model mass Mm in Equation (10) will be dependent on θ. The measured mode shapes are massnormalized with respect to the reduced model mass Mm. Therefore the measured modal flexibility matrix, which is constructed by measured modal frequencies and massnormalized mode shapes, will be dependent on θ. Most probable value and uncertainty In the case when sufficient amount of modal flexibility data sets are available and the incompleteness of the modal parameters is not significant, the model updating problem is usually globally identifiable. By maximizing the posterior PDF, the most probable stiffness parameter vector can be determined. Instead of maximizing the posterior PDF, one can equivalently minimize the objective function J ( ) ln p( D, M ) to obtain the optimal (most 180 . centered at the optimal (most probable) parameters with covariance matrix equal to the inverse of the Hessian of parameters, i.e., (27) probable) stiffness parameter vector (29) the function J ( ) ln p( D) calculated at the optimal 1 p( fˆt , M ) c1 exp [ fˆt f ( )]T 1[ fˆt f ( )] 2 5.2 1 Nt ˆ [ ft f ( )]T 1[ fˆt f ( )] 2 t 1 This can be done by using the built-in function “fminsearch” or “fminunc” in MATLAB. Using Laplace’s asymptotic approximation, the posterior PDF can be well approximated by a Gaussian distribution (26) where f is the averaged experimental modal flexibility vector. For the tth set of modal flexibility vector data, the likelihood function can be expressed as function of the model parameters θ, J ( ) [ H ( )]1 [J ( )T ]1 . This can be done by using finite difference. It should be noted that if the model is locally identifiable or unidentifiable, the approach presented above is not applicable and sampling-based approaches may be helpful to find the set of all optimal parameters. 6 SUMMARY OF PROCEDURES THE MODEL UPDATING In the complete measurements case, the model updating flow chart is shown in Figure 1. The procedures are summarized as follows: (1) A sequence of modal frequencies and complete mode shapes data are identified and the mode shapes are mass normalized. (2) The modal flexibility matrices are constructed from the modal frequencies and mass-normalized mode shapes. Then the modal flexibility vectors are obtained by vectorization of the lower triangular portion of the modal flexibility matrices. The covariance matrix of the modal flexibility vectors is also calculated. (3) An initial estimate of the model parameters is obtained by using the deterministic model updating technique described in Section 4. (4) The modal flexibility vectors and their covariance matrix are incorporated in a Bayesian updating formula. The most probable values of the model parameters are obtained by maximizing the posterior PDF of the model parameters, where the initial guess in this optimization problem is set to be equal to the initial estimate of the model parameters obtained in the deterministic updating stage. The covariance matrix of the model parameters is calculated at the most probable values by finite differences. Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 by vectorization of the lower triangular portion of the modal flexibility matrices. (3) An initial estimate of the model parameters is obtained by using the deterministic model updating technique described in Section 4. (4) The modal flexibility vectors and their covariance matrix are dependent on the updated model parameters when the mode shapes are incomplete. The modal frequencies and incomplete mode shapes data sets are directly incorporated in a Bayesian updating formula. The most probable values of the model parameters are obtained by maximizing the posterior PDF of the model parameters, where the initial guess in this optimization problem is set to be equal to the initial estimate of the model parameters obtained in the deterministic updating stage. The covariance matrix of the model parameters is calculated at the most probable values by finite differences. 7 Figure 1 Model updating with complete modal flexibility data. 7.1 ILLUSTRATION EXAMPLE Model updating In this example an eight-story shear building is considered. It is assumed that this building has uniformly distributed floor mass and uniform inter-storey stiffness. The mass per floor is taken to be 2×104 kg, the nominal value of the inter-storey stiffness of each floor is 15×106 N/m. The real inter-storey stiffness of the second and fourth floor is assumed to have a reduction due to damage of 20% and 40%, respectively. For the simulated modal data, samples of zero-mean Gaussian noise with covariance matrix were added to the exact modal frequencies and mode shapes. The covariance matrix is assumed to be diagonal with the variances corresponding to 1% coefficient of variation for both the modal frequencies and mode shapes for all modes, a reasonable value for typical modal testing. The nominal substructure stiffness matrices are given by 15 106 K1 071 017 077 (30) for the first storey and Figure 2 Model updating with incomplete modal flexibility data In the case of incomplete measurements, the model updating flow chart is shown in Figure 2. The procedures are summarized as follows: (1) A sequence of modal frequencies and incomplete mode shapes data are identified and the mode shapes are initially mass normalized with respect to the nominal reduced model mass. (2) The initial modal flexibility matrices are constructed from the modal frequencies and initial mass-normalized mode shapes. Then the initial modal flexibility vectors are obtained 0(l 2)8 0 6 6 1(l 2) 15 10 15 10 01(l 2) 15 106 15 106 0(8l )8 01(8l ) 01(8l ) (31) for other stories, i.e., l = 2, 3,…, 8. For the first case, complete measurements are considered, i.e., all the DOFs are assumed to be measured and the first four modes are used to construct the modal flexibility matrix for model updating. Figure 3 shows the identified most probable values of the stiffness parameters. From the figure, we can see that the second floor and the fourth floor have damages of about 20% and 40%, respectively. Table 1 shows the identified most probable values, the calculated standard deviations (SD), coefficient of variation (CV) for each parameter, and the value of a ‘normalized distance’ (ND) for 181 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 each parameter. The ‘normalized distance’ represents the absolute value of difference between the identified value and actual value, normalized with respect to the corresponding calculated standard deviation. From the identified results, we can see that the proposed method can update the model successfully. For the second case, incomplete measurements are considered such that the 1st, 3rd, 5th, 7th and 8th DOFs are measured, and the first four modes are used to construct the modal flexibility matrix for model updating. In order to check the efficacy of the model reduction technique for mode shape mass-normalization, the first four mode shapes of the nominal model are mass-normalized with respect to full model mass and reduced model mass, respectively. The first four mode shapes of the nominal model which are mass-normalized with respect to full model mass are shown in Table 2, while those mass-normalized with respect to reduced model mass are shown in Table 3. Comparing Table 2 with Table 3, we can see that they are identical, which validates the efficacy of the model reduction technique for mode shape massnormalization. Figure 4 shows the identified most probable values of the stiffness parameters. From the figure, we can see that the second floor and the fourth floor have damages of about 20% and 40%, respectively. Table 4 shows the identified most probable values, the calculated standard deviations (SD), coefficient of variation (CV) for each parameter, and the value of a ‘normalized distance’ (ND) for each parameter. From the identified results, we can see that the proposed method can update the model successfully using incomplete measurements. Comparing the standard deviations identified from complete and incomplete measurement data as shown in Table 1 and 4 respectively, we can see that the standard deviations identified from incomplete measurement data is larger, which indicates that larger uncertainties exist in the case of incomplete data. Figure 4 Identified most probable values of stiffness parameters (incomplete measurement) Table 1 Identification results of damaged structure (complete measurement) Parameter θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 true 0 -0.2 0 -0.4 0 0 0 0 identified -0.0008 -0.1974 0.0024 -0.3999 0.0021 0.0040 0.0028 -0.0002 SD 0.0010 0.0012 0.0011 0.0007 0.0011 0.0019 0.0013 0.0011 CV 1.1935 0.0058 0.4477 0.0017 0.5347 0.4749 0.4756 5.1308 ND 0.8378 2.2761 2.2336 0.1683 1.8701 2.1059 2.1025 0.1949 Table 2 Mass-normalized mode shapes with respect to full model mass 1st DOF 1st mode 2nd mode 3rd mode 4th mode -0.0006 -0.0018 -0.0027 0.0033 rd -0.0018 -0.0034 -0.0012 -0.0023 th -0.0027 -0.0012 0.0034 0.0006 th -0.0033 0.0023 -0.0006 0.0012 th -0.0034 0.0033 -0.0031 -0.0027 3 DOF 5 DOF 7 DOF 8 DOF Table 3 Mass-normalized mode shapes with respect to reduced model mass Figure 3 Identified most probable values of stiffness parameters (complete measurement) 182 1st mode 2nd mode 3rd mode 4th mode 1st DOF -0.0006 -0.0018 -0.0027 0.0033 3rd DOF -0.0018 -0.0034 -0.0012 -0.0023 th -0.0027 -0.0012 0.0034 0.0006 th 7 DOF -0.0033 0.0023 -0.0006 0.0012 8th DOF -0.0034 0.0033 -0.0031 -0.0027 5 DOF Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Table 4 Identification results of damaged structure (incomplete measurement) Parameter θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8 7.2 true 0 -0.2 0 -0.4 0 0 0 0 identified -0.0021 -0.1983 -0.0008 -0.395 -0.0129 -0.0093 0.0029 0.0015 SD 0.0013 0.0034 0.0049 0.0023 0.0052 0.0064 0.0065 0.0025 CV 0.6182 0.0174 5.9819 0.0059 0.4026 0.684 2.2631 1.6859 ND 1.6175 0.4877 0.1672 2.1464 2.4837 1.4619 0.4419 0.5932 Structural health monitoring If the Bayesian model updating approach is used to update the probability density function (PDF) of the stiffness parameters of the structural model based on the measured modal flexibility data corresponding to the undamaged and possibly damaged states, possible structural damage due to stiffness loss can be detected. The detection of damage is based on the probability that a sub-structure stiffness parameter has a fractional decrease compared to its value corresponding to the undamaged structure. In order to portray the damage, the identified most probable values and their calculated standard deviations for the stiffness parameters are used to find the probability that a given stiffness parameter n has been reduced by certain fraction d compared to its value corresponding to the undamaged state. An asymptotic Gaussian approximation is used for the integrals involved to give: Pndm (d ) P((1 npd ) (1 d )(1 nud )) (1 d )(1 ud ) (1 pd ) n n (1 d ) 2 ( ud ) 2 ( pd ) 2 n n where (32) () is the standard Gaussian cumulative distribution function; nud and npd denote the most probable values of the stiffness parameters for the undamaged and (possibly) damaged structure, respectively; nud and npd are the corresponding standard deviations of the stiffness parameters. The structural model used is the same as before. Incomplete measurements are considered such that the 1st, 3rd, 5th, 7th and 8th DOFs are measured, and the first four modes are used to construct the modal flexibility matrix for model updating. The stiffness parameters n (n=1,2,…,8) for the undamaged nd Figure 5 Probability of damage curves 8 ACKNOWLEDGMENTS This research has been supported by the Hong Kong Research Grants Council under grants 613511, 613412 and 613713. These supports are gratefully acknowledged. REFERENCES [1] th structure are assumed to be zero, while the 2 and 4 stories are assumed to be damaged with a stiffness loss of 20% and 40% respectively. The probabilities of damage for the eight stories are shown in Figure 5. It can be clearly seen that the second storey and the fourth storey have damage with probability almost unity and the mean of the damage percentage is 20% and 40%, respectively. CONCLUSION A Bayesian structural model updating methodology based on modal flexibility is presented with application to structural health monitoring. The proposed method is based on modal flexibility matrix which can be constructed from identified modal frequencies and mode shapes. The most probable values of the model stiffness parameters are obtained by maximizing the posterior PDF of the model parameters. The associated uncertainties are also quantified by calculating the covariance matrix of the model parameters at the most probable values. In order to facilitate the optimization in Bayesian updating, a deterministic initial updating stage is added before the Bayesian updating stage with the purpose of identifying an initial estimate of the model stiffness parameters. Furthermore, the SEREP model reduction technique is adopted to cope with the problem of mismatch between the DOFs of the FE model and those of the experimental model. 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