# Dynamic-based F.E. model updating to evaluate damage in ... ?· (Bendat & Piersol 1993) and the Frequency…

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<ul><li><p>Structural Analysis of Historical Constructions - Modena, Loureno & Roca (eds) 2005 Taylor & Francis Group, London, ISBN 04 1536 379 9 </p><p>Dynamic-based F.E. model updating to evaluate damage in masonry towers </p><p>C. Gentile & A. Saisi Department ofStrue/ural Engineering, Po/iteenieo of Milan, Milan, Ita/y </p><p>ABSTRACT: The paper presents the experimental and analytical modal analysis of a masonry bell-tower, dating back to the XVII century. The tower, about 74 m high, is characterised by the presence of major cracks on the Westem and Eastern load-bearing walls. The field test was carried out by ambient vibration testing; both the classical Peak Pieking spectral technique and the more advanced Frequeney Domain Decomposition procedure were used to extract the modal parameters (natural frequencies and mode shapes) from ambient vibration data. [n the theoretical study, vibration modes were determined by using a 3D finite element model. The experimental data were first used to verify the main assumptions used in the models through rough comparison of measured and predicted modal parameters; furthermore, some structural parameters of the model were updated in order to enhance the match between theoretical and experimental modal parameters. </p><p>rNTRODUCT[ON </p><p>[nvestigations on the structural safety of ancient masonry towers have recently become of increasing concern , probably as a consequence of some dra-matic events which occurred like the sudden collapse of the Civic Tower in Pavia in 1989 (Binda et aI. 1992, Binda et aI. 1995). Generally, such investigations include: </p><p>a. accurate survey ofthe crack pattern and geometric assessment; </p><p>b. non-destructive and slightly destructive tests, like flat-jack tests or sonic pulse velocity tests; </p><p>c. various other laboratory tests on cored samples; d. finite element modelling and theoretical analysis. </p><p>[n the paper, the results of the dynamic-based assessment of an historic masonry bell-tower, adja-cent to the Cathedral of Monza (a town about 20 km far from Milan, Italy), are presented and discussed. Dynamic-based assessment of a structure generally involves the comparison between the experimental modal parameters identified during full-scale tests and the predictions of finite elements analysis, as it is schematically shown in the flow-chart of Fig. I . The figure clearly outlines the ma in steps of a dynamic-based assessment procedure: </p><p>I . Full-scale dynamic testing; 2. Experimental modal analysis (EMA), i.e. the </p><p>extraction of modal parameters (natural frequen-cies and mode shapes) from experimental data; </p><p>3. Finite element analysis (FEA) and correlation with the experimental results; </p><p>4. Model updating (Mottershead & Friswell 1993). </p><p>Full-scale dynamic tests were carried out to com-plement an extensive experimental program planned to assess the structural condition ofthe tower since the West and East sides of the building exhibited wide, passing-through and potentially dangerous vertical cracks. </p><p>The EMA was carried out in the frequency domain by using the classical Peak Pieking spectral techniques (Bendat & Piersol 1993) and the Frequency Domain Decomposition procedure (Brincker et aI. 200 I). The fundamental mode, with a natural frequency of about 0.59 Hz, involves dominant bending in the E-W direction with significant bending participation in the opposite N-S direction as well. Notwithstanding the nearly symmetric shape, the identified modes of the system generally show coupled motion in the two main E-W and N-S directions; thus, the EMA suggests either a significant interaction between the bell-tower and the Cathedral or a non-symmetric stiffness dis-tribution (as the one expected basing on the crack distribution). </p><p>In the theoretical study, vibration modes were deter-mined by using a 3D finite element model. Exper-imental modal data were then used to verify the main assumptions adopted in formulating the model and to adjust some uncertain structural parameters. The updated model, characterised by relatively low </p><p>439 </p></li><li><p>MODEL UPDATING NO </p><p>Figure I. Dynamic-based assessment of a structure. </p><p>stiffness ratios in the damaged regions of the tower, exhibits good agreement in both frequencies and mode shapes (at the measurement locations) for ali identified modes. </p><p>Since the tower is currently subjected to a repair intervention, a long term goal of this research is to repeat the dynamic testing after the strengthening in order to investigate the correlation between repair and changes in the modal parameters of the structure. </p><p>2 THE BELL-TOWER IN MONZA </p><p>2.1 Damage description </p><p>The bell-tower ofthe Monza's Cathedral (Fig. 2) was built between 1592 and 1605, probably according to the design of Pellegrino Tibaldi. Since the erection required only 14 years, a general uniformity of the construction techniques and materiais characterises the tower. </p><p>The load-bearing walls ofthe tower, 74 m high and 1.40 m th ick, were made with solid masonry bricks and showed passing-through, large and potential1y danger-ous vertical cracks especially on the West and East sides (Fig. 3). These cracks were certainly present before 1927 (when a rough monitoring of the cracks started) and are slowly but continuously opening. Other vertical and very thin cracks can be observed </p><p>Min. J ? </p><p>FULL-SCALE DYNAMIC TESTING </p><p>EMA Responses (natura l frequencies </p><p>& mode shapes) </p><p>YES OPTIMAL MODEL </p><p>Figure 2. The bell-tower and the Cathedral of Monza. </p><p>440 </p></li><li><p>__ Passing through cracks </p><p>Deep cracks </p><p>O", </p><p>.... </p><p>o </p><p>WESTSIDE </p><p>D I, ' </p><p>NORTH SIDE EASTSIDE </p><p>ENTRANCE </p><p>SOUTH SIDE </p><p>Figure 3. Crack pattern on the externaI walls ofthe Tower (dimensions in metres). </p><p>mainly on the inner faces of the bearing walls; these further cracks are widespread along the four sides of the tower and deeper at the sides of the entrance were the stresses are more concentrated. The observed crack pattern is present approximately from a height of 11 .0 m up to 23.0 m. Since the cracks have developed slowly along the years, a possible time dependent behaviour of the material can be supposed due to the heavy dead load, coupled to temperature variations and wind actions (Binda et al. 1995). </p><p>2.2 On-site invesligalions </p><p>The complete results of on-site investigations is reported in Binda & Poggi 1997, Binda et al. 2000. </p><p>First, an accurate geometric survey of the struc-ture was carried out and included the analysis of the crack patterns and distribution; cracks were surveyed visually and photographically and reported on plans, prospects and sections. Successively, tlat-jack tests were performed in selected points to directly estimate the stress levei caused by the dead load. Some double </p><p>441 </p><p>tlat-jack tests were also carried out to check the stress-strain behaviour of the masonry under compression; specifically, the Young 's modulus was generally rang-ing between 985 and 1380 N/mm2 while a Poisson ratio ofO.07-O.20 was detected. </p><p>Furthermore, a first series of dynamic tests using four servo-accelerometers was carried out in 1995 to evaluate the natural frequencies of the tower and a structural model was developed by using the results of the test on the materiaIs (Binda & Poggi 1997). </p><p>3 FULL-SCALE DYNAMIC TESTrNG </p><p>Extensive full-scale dynamic tests were carried out at the beginning of July 2001 to measure the dynamic response of the Tower at 20 different locations, with the excitation being associated to environrnentalloads and to the bell ringing. Figure 4 shows a schematic representation of the sensor layout. </p><p>WR-71 piezoelectric sensors (Fig. 5) were used during the tests; these sensors allowed acceleration </p></li><li><p>48.42 </p><p>5.70 </p><p>5.70 </p><p>5.70 </p><p>5.70 </p><p>3 .80 </p><p>5.70 </p><p>5 .70 </p><p>5.35 </p><p>5 .07 </p><p>Figure 4. Sensor locations. </p><p>Figure 5. WR-71 accelerometer. </p><p>or velocity responses to be recorded. Two-conductor cables connected the accelerometers to a computer workstation with a data acquisition board for AlD and DI A conversion of the transducer signals and storage of digital data. </p><p>Ambient vibration response (in terms ofboth accel-eration and velocity) was acquired in about 38 minute records per channel at a sample rate of 200 Hz to provide good waveform definition. </p><p>Due to the low leveI of ambient excitation that existed during the tests, the maximum recorded veloc-ity ranges up to about 0.15 mm/s. </p><p>4 MODAL IDENTIFICATION PROCEDURES </p><p>The extraction of modal parameters from ambient vibration data was carried out by using two dif-ferent output-only procedures: Peak Picking method (PP, Bendat & Piersol 1993) and Frequency Domain Decomposition (FDD, Brincker et a!. 2001). Both methods are based on the evaluation of the spectral matrix (i.e. the matrix of cross-spectral densities) in the frequency domain: </p><p>G(f) = E[A(f)A H (f)] (1) </p><p>where the vector A(f) collects the acceleration responses in the frequency domain, superscript fi denotes complex conjugate transpose matrix and E denotes expected value. The diagonal terms of the matrix G(f) are the (real valued) auto-spectral den-sities (ASD) while the other terms are the (complex) cross-spectral densities (CSD): </p><p>(2a) </p><p>(2b) </p><p>where the superscript * denotes complex conjugate. Both ASDs and CSDs were estimated from recor-</p><p>ded data samples by using the modified periodogram method (Welch 1967); according to this approach an average is made over each recorded signal, divided into M frames of2n samples, where windowing and over-lapping is applied. In the present application, smooth-ing is performed by 8 I 92-points Hanning-windowed periodograms that are transformed and averaged with 50% overlapping; since I">t = 0.005 s, the resulting frequency resolution is 1/(8192 x 0.005) "" 0.0244 Hz. </p><p>4.1 Peak picking </p><p>The more traditional approach to estimate the moda I parameters of a structure (Bendat & Piersol 1993) is often called Peak Picking method. The method leads to reliable results provided that the basic assump-tions of 10w damping and well-separated modes are satisfied. In fact, for a lightly damped structure sub-jected to a white-noise random excitation, both ASDs and CSDs reach a local maximum at the frequencies </p><p>442 </p></li><li><p>corresponding to the system normal modes; hence, for well-separated modes, the spectral matrix can be approximated in the neighborhood of a resonant frequency f,. by: </p><p>(3) </p><p>where ar depends on the damping ratio, the natu-ral frequency, the modal participation factor and the excitation spectra. Eq. (3) highlights that: </p><p>I. each row or column ofthe spectral matrix at a reso-nant frequencyj;. can be considered as an estima te of the mode shape ifJr at that frequency; </p><p>2. the square-root ofthe diagonal terms ofthe spectral matrix at a resonant frequency f,. can be consid-ered as an estimate of the mode shape ifJ,. at that frequency. </p><p>In the present application of the PP method, nat-ural frequencies were identified from resonant peaks in the ASDs and in the amplitude of CSDs, for which the cross-spectral phases are O or ][ . The mode shapes were obtained from the amplitude of square-root ASD curves while CSD phases were used to determine directions of reI ative motion. </p><p>Drawbacks of the PP method (Abdel-Ghaffar and Houner 1978) are related to the difficulties in identify-ing closely spaced modes (beca use of spectral overlap) and damping ratios. </p><p>4.2 Frequency domain decomposition </p><p>The FDD approach is based on the singular value decomposition (se e e.g. Golub & Van Loan 1996) of the spectral matrix at each frequency: </p><p>(4) </p><p>where the diagonal matrix :E collects the real positive singular values in descending order and U is a complex matrix containing the singular vectors as columns. </p><p>[f only one mode is important at a certain frequency f,., the spectral matrix can be approximated by a rank-one matrix and can be decomposed as: </p><p>(5) </p><p>By comparing eq. (5) with eq . (3), it is evident that the first singular value a I (f) at each frequency repre-sents the strength ofthe dominating vibration mode at that frequency while the corresponding singular vector 11 I (f) contains the mode shape; the successive sin-gular values contain either noise or modes c10se to a strong dominating one. The FDD is a rather simple </p><p>procedure that represents an improvement of the PP since: </p><p>I. the singular value decomposition is at least an effective method to smooth the spectral matrix; </p><p>2. the evaluation of mode shapes is automatic and significantly easier than in the PP; </p><p>3. in case of c10sely spaced modes around a certain frequency, every singular vector corresponding to a non-negligible singular value can be considered as a mode shapes estimate. </p><p>4.3 Mode shapes correlation </p><p>Once the modal identification phase was completed, the two sets of mode shapes resulting from the appli-cation of PP and FDD were compared by using the well-known Modal Assurance Criterion (MAC, Allemang & Brown 1982) and the Normalized Modal Difference (NMD, Waters 1995). </p><p>The MAC is probably the most commonly used pro-cedure to correlate two sets of mode shape vectors and is a coefficient analogous to the correlation coefficient in stat istics. The MAC ranges from O to I; a value of I implies perfect correlation of the two mode shape vectors while a value c10se to O indicates uncorrelated (orthogonal) vectors. In general, a MAC value greater than 0.80 is considered a good match while a MAC value less than 0.40 is considered a poor match. </p><p>The NMD is related to the MAC by the following (Maya and Silva 1997): </p><p>1- MAC(~A.k ' ~B,j) </p><p>MAC(~A ,k'~B,j) (6) </p><p>In practice, the NMD is a c10se estimate ofthe average difference between the components of the two vec-tors ifJA ,k> ifJBJ and is much more sensitive to mode shape differences than the MAC. For example, a MAC of 0.950 implies a NMD of 0.2294, meaning that the components ofvectors ifJA ,k and ifJB j differ on average of 22.94%. The NMD is not bounded by unity; thus the comparison is more difficult for weakly correlated modes but is more discriminating when two modes are highly correlated, as it happens in the present application. </p><p>The MAC and the NMD were also used to correlate the results of FEA and EMA. </p><p>5 EXPERIMENTAL RESULTS </p><p>Five vibration modes were identified from ambient vibration data in the frequency range of O- 10Hz. </p><p>The results of the PP method in terms of natural frequencies can be demonstrated through the spectral </p><p>443 </p></li><li><p>(a) 0.09 </p><p>N ~ ~ 0.06 </p><p>E '=' o </p><p>~ ~ ~ 0.03 </p><p>0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 </p><p>frequency (Hz) </p><p>(b) 140 </p><p>N ~ 105 </p><p>~ ~ 70 :> (JJ </p><p>35 I A J~ J \ o </p><p>QOO 1m ~OO 3~ ~OO ~OO 6~ ~OO ~OO frequency (Hz) </p><p>Figure 6. (a) Autospectra of the response from different points of the tower; (b) First (largest) singular value of the spectral matrix. </p><p>Table I . Modal parameters identified during ambient vi bration tests. </p><p>Mode No. ModeType </p><p>I Bending mode in E-W/N-S di rection 2 Bending mo de in N-S/E-W direction 3 Torsional mo de 4 Bending mode in E-W/N-S di rection 5 Bendi ng mode in E-W/N-S direction </p><p>plots of Fig. 6(a), showing the (velocity) ASDs from different locations of the tower. Figure 6(a) clearly shows resonant...</p></li></ul>

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