Modal identification of heritage structures: Molise s bell towers

Transcription

1 Experimental Vibration Analysis for Civil Engineering Structures EVACES 09 Modal identification of heritage structures: Molise s bell towers C. Rainieri, C. Laorenza & G. Fabbrocino Structural and Geotechnical Dynamic Laboratory StreGa, University of Molise, Termoli, Italy SAVA Department, University of Molise, Campobasso, Italy ABSTRACT: An effective protection of construction at seismic risk can be reached by increasing the knowledge of the dynamic behaviour of structures, particularly in the case of historical structures. Unique structural techniques and uncertainties affecting geometry and material properties make accurate structural analyses of heritage structures and assessment of their effective behaviour even in operational conditions difficult. Thus an effective evaluation of modal properties of historical structures is crucial in order to assess the structural performances under extreme loads, such as earthquakes. Output-only techniques for dynamic identification are preferred in the case of heritage constructions, since artificial excitation often exhibits problems of test execution and input control, conversely environmental actions are always present. In addition, tests are less expensive and faster compared with traditional EMA and imply a minimum interference with the normal use of the structure. This paper deals with the structural identification of a number of masonry bell towers in Molise region (Southern Italy), some of which were seriously damaged after the Molise earthquake in In particular, the main results of an experimental campaign based on dynamic tests are reviewed, and the results obtained by applying different output-only modal analysis methods are compared. 1. INTRODUCTION The development, in recent years, of a number of methodologies for seismic assessment of structures is a consequence of the increasing awareness about seismic vulnerability and risk of existing structures. Identification of the main weakness and deficiencies of such systems and design of interventions for risk mitigation is a complex task, in particular for historical or heritage structures (Cardoso et al., 2005; Ivorra & Pallarès, 2006; Betti & Vignoli, 2008; Turer & Boz, 2008). This is mainly due to a number of relevant aspects: characterization of geometry, material properties, interventions carried out on the structure throughout its life, existing damages (Lourenço, 2003). Thus, reliable numerical analyses are difficult, due to the complexity of the global structural system, and the use of standard structural schemes to study historical buildings could provide meaningless results. As a consequence, specific modeling and analysis strategies as well as appropriate experimental campaigns are needed. In this framework, experimental modal analysis is becoming more and more relevant as a tool for investigation of the dynamic behaviour of important structures from the historical or architectural point of view (Direttiva P.C.M., 2007). The knowledge of the modal properties of historical structures is very important, in particular for the evaluation of the structural performance under severe and extreme loads, such as earthquakes (Gentile, 2005). In the last thirty years several techniques aimed at the experimental evaluation of the dynamic characteristics of structures have been developed. Beside traditional techniques based on the knowledge of the input source, increased attention has been paid to techniques for modal parameters identification based on ambient vibrations which, among the rest, allow the evaluation of the dynamic properties of 161

2 EVACES 09 Vibrations of large structures a structure in its actual service conditions (Cunha & Caetano, 2005) without any specific external excitation. As historical structures are concerned, output-only techniques are preferred (Gentile, 2005), since artificial excitation often exhibits problems of test execution and input control while the environmental loads are always present. In addition, tests are less expensive and faster with respect to traditional experimental modal analysis and imply a minimum interference with the normal use of the structure (Mohanty, 2005). Identified modal parameters, representative of the structural behaviour in operational conditions, can be used to validate or update finite element models. Moreover, they can be used to evaluate possible changes in the modal parameters which can be correlated with structural modifications or damage. Finally, combination between numerical models and experimental measures offers interesting opportunities in the fields of vibration and seismic protection of strategic or historical constructions. In fact, updated analytical models can be used for an efficient evaluation of the seismic risk of the structure itself. The present paper deals with use of Operational Modal Analysis for the evaluation of the modal parameters of a number of bell towers belonging to the cultural heritage of Molise region in Southern Italy. Most of them suffered serious damages after the Molise earthquake in 2002 (Cifani et al., 2005). The need to design appropriate restoration and seismic upgrading interventions, taking into account the valuable characteristics of such structures, and, therefore, the need of improved knowledge of their structural characteristics suggested to carry out dynamic identification tests. The present paper is focussed on this specific aspect and is aimed at the discussion of the methods and of the results of an experimental assessment of the dynamic properties of a number of bell towers based on environmental vibrations. 2. DATA PROCESSING METHODS Modal parameter estimation in output-only conditions has been carried out according to the Stochastic Subspace Identification (SSI) (Van Overschee & De Moor, 1996; Peeters, 2000), the Enhanced Frequency Domain Decomposition (EFDD) (Brincker et al. 2000) and the Second Order Blind Identification (Poncelet et al., 2007). As (Enhanced) Frequency Domain Decomposition technique is concerned, it is an extension of the Basic Frequency Domain method, often called Peak-picking (Bendat & Piersol 1993). It is based on the Singular Value Decomposition (SVD) of the Power Spectral Density (PSD) matrix, which has been previously estimated directly from the raw data at discrete frequencies ω=ω i : G H yy j i U i S i U i (1) where the matrix [U] i is a unitary matrix holding the singular vector u ij and [S] i is a diagonal matrix holding the scalar singular values s ij. Near a peak corresponding to the k th mode in the spectrum, this mode will be dominant. If only the k th mode is dominant, there will be one term in eq. (1) and the PSD matrix approximates to a rank one matrix as: G yy 1 H ji siu i u i i k 1 (2) In such case, the first singular vector u is an estimate of the mode shape: i1 ˆ u i1 (3) and the corresponding singular value belongs to the Auto Power Spectral Density function of the corresponding Single Degree Of Freedom (SDOF) system. In case of repeated modes, the PSD matrix rank is equal to the number of multiplicity of the modes. The Auto Power Spectral Density function of the corresponding SDOF system is identified around the peak of the singular value plot by comparing the mode shape estimate ˆ with the singular vectors associated to the frequency lines around the peak: 162

3 Modal identification of heritage structures: Molise s bell towers every line characterized by a singular vector which gives a MAC value (Allemang & Brown, 1982) with ˆ higher than a user-defined MAC Rejection Level belongs to the SDOF PSD function. This equivalent SDOF PSD function is used, according to EFDD algorithm, for natural frequency and damping estimations. The former are independent upon the frequency resolution of the spectra computed by the Fast Fourier Transform (FFT) algorithm. In fact, the SDOF PSD function is transferred back to time domain through Inverse FFT, so that an approximated correlation function of the equivalent SDOF system is obtained. From the free decay function of the SDOF system, the damping ratio can be calculated by the logarithmic decrement technique. A similar procedure is adopted in order to extract natural frequencies, performing a linear regression on the zero crossing times of the equivalent SDOF system correlation function and taking into account the relation between damped and undamped natural frequency. The SVD of the PSD matrix, which is the core of the FDD algorithm, allows to overcome the typical drawbacks of the BFD technique. The SVD, in fact, is a standard linear algebra tool for estimating the rank of a matrix (the number of non-zero singular values is the rank). Its application in this context allows to solve the problem of mode multiplicity. In this case, every singular vector corresponding to a non-zero singular value yields a mode shape estimate, if the mode shapes are orthogonal each other. However, this is not always true: in such a case, the first singular vector is still a good estimate of a mode shape, but this is not true for the other. Covariance-Driven (Cov-SSI) and Data-Driven (DD-SSI) Stochastic Subspace Identification methods are, instead, both time domain methods based on a state space description of the dynamic problem. In fact, the second order problem, stated by the differential equation of motion, is converted into two first order problems, defined by the so-called state equation and observation equation. Such equations, in the output-only case, can be written as follows: x k 1 A x k w k ; y k C x k v k (4) where x k xkt u k and y k are the sampled input and output, A is the discrete state matrix, output matrix, w is the process noise due to disturbances and model inaccuracies, is the discrete-time state vector yielding the sampled displacements and velocities, k C is the discrete v is the measurement noise due to sensor inaccuracy. These vector signals cannot both be measured. They are assumed to be zero mean Gaussian white noise processes with covariance matrices given by: E where E is the expected value operator, pq w p T T Q S wq vq T pq v S R p pq is the Kronecker delta (if p=q then 1, otherwise 0 ), p and q are two arbitrary time instants. In the stochastic framework which characterizes Operational Modal Analysis, due to the lack of information about the input, it is implicitly modelled by the noise terms w k and v k. The white noise assumption about w k and v k cannot be omitted for the proof of this class of identification methods (see also Van Overschee & De Moor, 1996). If this assumption is violated, that is to say the input includes white noise and some additional dominant frequency components, such components will appear as poles of the state matrix [A] and cannot be separated from the eigenfrequencies of the system. Cov-SSI and DD-SSI are two algorithms for estimation of system matrices and, therefore, of eigenproperties. The main difference is related to the fact that Cov-SSI works on output correlations while DD-SSI works directly on raw data. Second Order Blind Identification (SOBI) is a Blind Source Separation (BSS) technique for signal processing and data analysis that, given a series of observed signals, aims at recovering the underlying sources under the assumption of their mutual independence. In the case of linear and static mixtures, the noisy model can be expressed in matrix form as: xt A st t (6) pq k (5) 163

4 EVACES 09 Vibrations of large structures where [x(t)] represents the recorded data, [s(t)] are the source signals, [σ(t)] is the noise corrupting the data and [A] is referred to as the mixing matrix. The aim of BSS is, therefore, to recover the sources [s(t)] from their observed mixtures [x(t)], knowing very little about the mixing matrix [A] and making general assumptions about the sources. Applicability of BSS to vibration data is proved by taking into account that the physical responses [x(t)] are related to the modal responses [q(t)] by the modal matrix [Φ]: xt qt (7) Comparison of equations (6) and (7) shows that the modal coordinates may, under given assumptions, act as virtual sources regardless of the number and type of the physical excitation forces (Poncelet et al., 2007). It is worth noting that no mathematical model is assumed to describe the process that produced the measured data. As a consequence, BSS techniques can be addressed as nonparametric procedures for modal identification since the mixing model is the only assumption. The SOBI algorithm finds components that approximately produce diagonal time-shifted covariance matrices. The main steps of the algorithm can be summarized as follows: observed data [x(t)] are centralized, removing the means value from each component of [x(t)], and whitened (basically through a Principal Component Analysis); whitened data are used to construct a number of time shifted covariance matrices; a numerical algorithm based on the Jacobi rotation technique (Cardoso & Souloumiac 1996) is then used to recover the mixing matrix; mode shapes are extracted directly from the mixing matrix [A], while natural frequencies and damping ratios are obtained through a Single Degree Of Freedom (SDOF) curve fitting of the sources. More details about the SOBI algorithm can be found elsewhere (Poncelet et al., 2007). It is worth noting here that the SOBI algorithm, when applied for output only modal identification, can be classified as a non-parametric two-step method. However, unlike many other two-step modal identification methods, in this case mode shapes are estimated before natural frequencies and damping ratios. Another important feature of SOBI is the limit on the number of identifiable modes. In fact, it is possible to identify as many sources as the number of measurement locations. As a consequence, it is possible to identify a number of modes lower or equal to the number of measurement locations. The above mentioned data processing algorithms have been implemented into software packages developed in LabView environment ( for more details, see (Rainieri, 2008; Rainieri et al., 2009). 3. MONTELONGO BELL TOWER Montelongo s church (Figure 1) was built in the 12 th Century and it underwent a number of interventions throughout its life. In particular, the bell tower was demolished and rebuilt in Currently, it shows a reinforced concrete structure with masonry infills. The dynamic response of the structure has been measured at eight different levels by placing ten couples of accelerometers at opposite corners of the tower, so that both translational and torsional modes could be observed. Accelerometers are of the Force Balance type (Kinemetrics EpiSensor ES- U2). They have a bandwidth (-3 db) of about 200 Hz (starting from DC) at 1 g and a high dynamic range (more than 140 db). A full scale range of 0.5 g and 20 V/g sensitivity have been adopted. Data have been acquired through the TrioGuard32 recorder, characterized by a 16-bit DSP. A record length of 3600 sec has been considered, with a sampling frequency of 100 Hz. Data processing in frequency domain has been carried out considering a Hanning window and a 66% overlap in spectrum computation. 164

5 Modal identification of heritage structures: Molise s bell towers Figure 1. Montelongo Church and Bell Tower Results of modal identification are summarised in Table 1. Table 1. Montelongo bell tower modal identification results Mode EFDD Cov-SSI SOBI f [Hz] ξ [%] f [Hz] ξ [%] f [Hz] ξ [%] I II III IV V Thus, modal identification results provided by the different methods show a good agreement. In Figure 2 the Singular Value plots provided by the EFDD method are shown, while in Figure 3 the complexity plots of the identified mode shapes are reported. It is easy to recognize that all modes are normal or nearly normal, since imaginary components are basically negligible. 165

6 EVACES 09 Vibrations of large structures Figure 2. Montelongo Bell Tower: Singular Value Plots Figure 3. Montelongo Bell Tower: Complexity Plots of the first three modes, from left to right (EFDD) 4. MONTORIO BELL TOWER Montorio s church (Figure 4) is another ancient structure made in the Middle Age. Its bell tower is characterized by a masonry structure, with a number of vaults at different levels linked by a stair. The dome was rebuilt some decades ago, and it is characterized by a reinforced concrete structure. The dynamic response of the structure has been measured at five different levels by placing ten couples of accelerometers at opposite corners of the tower, so that both translational and torsional modes could be observed. The measurement hardware is that one described in the previous section. A record length of 3600 sec has been considered, with a sampling frequency of 100 Hz. Data processing in frequency domain has been carried out by considering a Hanning window and a 66% overlap in spectrum computation. Results of modal identification are summarized in Table 2. Table 2. Montorio bell tower modal identification results Mode EFDD Cov-SSI SOBI f [Hz] ξ [%] f [Hz] ξ [%] f [Hz] ξ [%] I II III

7 Modal identification of heritage structures: Molise s bell towers Thus, modal identification results provided by the different methods show again a good agreement. In Figure 5 the spectra of the identified sources provided by the SOBI algorithm are shown, while in Figure 6 the complexity plots of the mode shapes provided by Cov-SSI are reported. All modes are normal or nearly normal due to negligible imaginary components. Figure 4. Montorio Church and Bell Tower Figure 5. Montorio Bell Tower: spectra of the identified sources (SOBI) Figure 6. Montorio Bell Tower: Complexity Plots of the first three modes, from left to right (Cov-SSI) 167

8 EVACES 09 Vibrations of large structures 5. RIPABOTTONI BELL TOWER Ripabottoni s church (Figure 7) is characterized by a masonry structure and it was built in the beginning of the 18-th Century. It is one of the most important heritage constructions in Molise region. The bell tower underwent some interventions throughout its life but the original structure has been preserved since now. The dynamic response of the structure has been measured at five different levels by placing ten couples of accelerometers at opposite corners of the tower, so that both translational and torsional modes could be observed. The measurement hardware is that one described in the previous section. A record length of 3600 sec has been considered, with a sampling frequency of 100 Hz. Data processing in frequency domain has been carried out by considering a Hanning window and a 66% overlap in spectrum computation. Figure 7. Ripabottoni Church and Bell Tower Results of modal identification are reported in Table 3. Table 3. Ripabottoni bell tower modal identification results Mode EFDD Cov-SSI SOBI f [Hz] ξ [%] f [Hz] ξ [%] f [Hz] ξ [%] I II III Thus, modal identification results provided by the different methods are in good agreement among them. In Figure 8 the stabilization diagram provided by the Cov-SSI method is shown, while in Figure 9 the complexity plots of the identified mode shapes are reported. Again, all modes are normal or nearly normal. 168

9 Modal identification of heritage structures: Molise s bell towers Figure 8. Ripabottoni Bell Tower: stabilization diagram Figure 9. Ripabottoni Bell Tower: Complexity Plots of the first three modes, from left to right (Cov-SSI) 6. CONCLUSIONS Identification of the main weakness and deficiencies of heritage constructions and design of interventions for risk mitigation is a complex task. In addition, reliable numerical analyses are difficult, due to the complexity of the global structural system, so that often standard approach to structural analysis of historical buildings may provide meaningless results. As a consequence, specific modeling and analysis strategies as well as appropriate experimental protocols are needed. In this paper the use of Operational Modal Analysis techniques for the evaluation of the modal parameters of a number of bell towers belonging to the cultural heritage of Molise region in Southern Italy has been described. Most of those bell towers suffered serious damages after the Molise earthquake in The need to design appropriate restoration and seismic upgrading interventions, taking into account the valuable characteristics of such structures, and, therefore, the need of improved knowledge of their structural characteristics suggested to carry out dynamic identification tests. Results of such tests obtained from different data processing methods have been reported and compared. The modal parameters provided by the different methods are in good agreement among them, pointing out the effectiveness and reliability of Operational Modal Analysis procedures for the identification of the most relevant dynamic parameters of this peculiar type of structures. 169

10 EVACES 09 Vibrations of large structures ACKNOWLEDGEMENTS The present work is part of an experimental campaign issued by the Diocese of Termoli and Larino, Molise (Italy) to assess relevant structural properties of bell towers damaged after Molise 2002 earthquake. Technical and logistic support by Dr. Eng. Enzo Palermo and Dr. Eng. Alberto Lemme is gratefully acknowledged. REFERENCES Allemang, R.J., Brown, D.L. (1982): A correlation coefficient for modal vector analysis, Proceedings of the 1st SEM International Modal Analysis Conference, Orlando, FL, USA. Bendat, J.S., Piersol, A.G. (1993): Engineering Applications of Correlation and Spectral Analysis, Second Edition, John Wiley & Sons, New York. Betti, M., Vignoli, A. (2008): Modelling and analysis of a Romanesque church under earthquake loading: Assessment of seismic resistance, Engineering Structures, 30, pp Brincker, R., Zhang, L., Andersen, P. (2000): Modal identification from ambient responses using frequency domain decomposition, Proceedings of the 18th SEM International Modal Analysis Conference, San Antonio, TX, USA. Cardoso, J.F., Souloumiac, A. (1996): Jacobi angles for simultaneous diagonalization, SIAM Journal of Matrix Analysis and Applications, 17, pp Cardoso, R., Lopes, M., Bento, R. (2005): Seismic evaluation of old masonry building. Part I: Method description and application to case-study, Engineering Structures, 27, pp Cifani G., Lemme A., Podestà S. (2005): Beni Monumentali e Terremoto, dall Emergenza alla Ricostruzione, DEI, Tipografia del Genio Civile (in Italian). Cunha, A., Caetano, E. (2005): From Input-Ouput to Output-Only Modal Identification of Civil Engineering Structures, Proceedings of the 1 st IOMAC Conference, Copenhagen, Denmark. Direttiva P.C.M. (2007): Direttiva del Presidente del Consiglio dei Ministri del 12/10/2007 per la valutazione e la riduzione del rischio sismico del patrimonio culturale con riferimento alle norme tecniche per le costruzioni, published at the G.U. 29/01/2008 n.24 (in Italian). Gentile, C. (2005): Operational Modal Analysis and Assessment of Historical Structures, Proceedings of the 1 st IOMAC Conference, Copenhagen, Denmark. Ivorra, S., Pallarès, F.J. (2006): Dynamic investigations on a masonry bell tower, Engineering Structures, 28, pp Lourenço, P.B. (2003): Computations on historic masonry structures, Progress in Structural Engineering and Materials, 4, pp Mohanty, P. (2005): Operational Modal Analysis in the presence of harmonic excitations, Ph.D. Thesis, Technische Universiteit Delft, The Netherlands. Peeters, B. (2000): System Identification and Damage Detection in Civil Engineering, Ph.D. Thesis. Katholieke Universiteit Leuven, Leuven, Belgium. Poncelet, F., Kerschen, G., Golinval, J.-C., Verhelst, D. (2007): Output-only modal analysis using blind source separation techniques, Mechanical Systems and Signal Processing, 21, pp Rainieri, C. (2008): Operational Modal Analysis for Seismic Protection of Structures, Ph.D. Thesis, University of Naples, Naples, Italy. Rainieri, C., Fabbrocino, G., Polito, T., Marulo, F. (2009): Blind Source Separation for acoustic and modal analysis: a comparative assessment, Proceedings of the 3 rd IOMAC Conference, Portonovo, Italy. Turer, A.,Boz, B. (2008): Computer modeling and seismic performance assessment of historic Aspendos theatre in Antalya, Turkey, Engineering Structures, 30, pp Van Overschee, P., De Moor, B. (1996): Subspace Identification for Linear Systems: Theory Implementation Applications, Kluwer Academic Publishers, Dordrecht. 170