Structural Analysis of Historical Constructions - Modena, Lourenço & Roca (eds) © 2005 Taylor & Francis Group, London, ISBN 04 1536 379 9

Dynamic-based F.E. model updating to evaluate damage in masonry towers

C. Gentile & A. Saisi Department ofStrue/ural Engineering, Po/iteenieo of Milan, Milan, Ita/y

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.

rNTRODUCT[ON

[nvestigations on the structural safety of ancient masonry towers have recently become of increasing concern , probably as a consequence of some dra­matic 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:

a. accurate survey ofthe crack pattern and geometric assessment;

b. non-destructive and slightly destructive tests, like flat-jack tests or sonic pulse velocity tests;

c. various other laboratory tests on cored samples; d. finite element modelling and theoretical analysis.

[n the paper, the results of the dynamic-based assessment of an historic masonry bell-tower, adja­cent 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 dynamic­based assessment procedure:

I . Full-scale dynamic testing; 2. Experimental modal analysis (EMA), i.e. the

extraction of modal parameters (natural frequen­cies and mode shapes) from experimental data;

3. Finite element analysis (FEA) and correlation with the experimental results;

4. Model updating (Mottershead & Friswell 1993).

Full-scale dynamic tests were carried out to com­plement 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.

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 dis­tribution (as the one expected basing on the crack distribution).

In the theoretical study, vibration modes were deter­mined by using a 3D finite element model. Exper­imental 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

439

MODEL UPDATING NO

Figure I. Dynamic-based assessment of a structure.

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.

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.

2 THE BELL-TOWER IN MONZA

2.1 Damage description

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.

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 danger­ous 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

Min. J ?

FULL-SCALE DYNAMIC TESTING

EMA Responses (natura l frequencies

& mode shapes)

YES OPTIMAL MODEL

Figure 2. The bell-tower and the Cathedral of Monza.

440

__ Passing through cracks

Deep cracks

O",

....

o

WESTSIDE

D I, '

NORTH SIDE EASTSIDE

ENTRANCE

SOUTH SIDE

Figure 3. Crack pattern on the externaI walls ofthe Tower (dimensions in metres).

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).

2.2 On-site invesligalions

The complete results of on-site investigations is reported in Binda & Poggi 1997, Binda et al. 2000.

First, an accurate geometric survey of the struc­ture 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

441

tlat-jack tests were also carried out to check the stress­strain behaviour of the masonry under compression; specifically, the Young 's modulus was generally rang­ing between 985 and 1380 N/mm2 while a Poisson ratio ofO.07-O.20 was detected.

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).

3 FULL-SCALE DYNAMIC TESTrNG

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.

WR-71 piezoelectric sensors (Fig. 5) were used during the tests; these sensors allowed acceleration

48.42

5.70

5.70

5.70

5.70

3 .80

5.70

5 .70

5.35

5 .07

Figure 4. Sensor locations.

Figure 5. WR-71 accelerometer.

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.

Ambient vibration response (in terms ofboth accel­eration and velocity) was acquired in about 38 minute records per channel at a sample rate of 200 Hz to provide good waveform definition.

Due to the low leveI of ambient excitation that existed during the tests, the maximum recorded veloc­ity ranges up to about 0.15 mm/s.

4 MODAL IDENTIFICATION PROCEDURES

The extraction of modal parameters from ambient vibration data was carried out by using two dif­ferent 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:

G(f) = E[A(f)A H (f)] (1)

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 den­sities (ASD) while the other terms are the (complex) cross-spectral densities (CSD):

(2a)

(2b)

where the superscript * denotes complex conjugate. Both ASDs and CSDs were estimated from recor­

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 over­lapping is applied. In the present application, smooth­ing 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.

4.1 Peak picking

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 assump­tions of 10w damping and well-separated modes are satisfied. In fact, for a lightly damped structure sub­jected to a white-noise random excitation, both ASDs and CSDs reach a local maximum at the frequencies

442

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:

(3)

where ar depends on the damping ratio, the natu­ral frequency, the modal participation factor and the excitation spectra. Eq. (3) highlights that:

I. each row or column ofthe spectral matrix at a reso­nant frequencyj;. can be considered as an estima te of the mode shape ifJr at that frequency;

2. the square-root ofthe diagonal terms ofthe spectral matrix at a resonant frequency f,. can be consid­ered as an estimate of the mode shape ifJ,. at that frequency.

In the present application of the PP method, nat­ural 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.

Drawbacks of the PP method (Abdel-Ghaffar and Houner 1978) are related to the difficulties in identify­ing closely spaced modes (beca use of spectral overlap) and damping ratios.

4.2 Frequency domain decomposition

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:

(4)

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.

[f only one mode is important at a certain frequency f,., the spectral matrix can be approximated by a rank­one matrix and can be decomposed as:

(5)

By comparing eq. (5) with eq . (3), it is evident that the first singular value a I (f) at each frequency repre­sents the strength ofthe dominating vibration mode at that frequency while the corresponding singular vector 11 I (f) contains the mode shape; the successive sin­gular values contain either noise or modes c10se to a strong dominating one. The FDD is a rather simple

procedure that represents an improvement of the PP since:

I. the singular value decomposition is at least an effective method to smooth the spectral matrix;

2. the evaluation of mode shapes is automatic and significantly easier than in the PP;

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.

4.3 Mode shapes correlation

Once the modal identification phase was completed, the two sets of mode shapes resulting from the appli­cation 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).

The MAC is probably the most commonly used pro­cedure 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.

The NMD is related to the MAC by the following (Maya and Silva 1997):

1- MAC(~A.k ' ~B,j)

MAC(~A ,k'~B,j) (6)

In practice, the NMD is a c10se estimate ofthe average difference between the components of the two vec­tors 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.

The MAC and the NMD were also used to correlate the results of FEA and EMA.

5 EXPERIMENTAL RESULTS

Five vibration modes were identified from ambient vibration data in the frequency range of O- 10Hz.

The results of the PP method in terms of natural frequencies can be demonstrated through the spectral

443

(a) 0.09

N ~ ~ 0.06

E '=' o

~ ~ ~ 0.03

0.00 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00

frequency (Hz)

(b) 140

N ~ 105

~ ~ 70

:> (JJ

35 I A J~ J \ o

QOO 1m ~OO 3~ ~OO ~OO 6~ ~OO ~OO

frequency (Hz)

Figure 6. (a) Autospectra of the response from different points of the tower; (b) First (largest) singular value of the spectral matrix.

Table I . Modal parameters identified during ambient vi bration tests.

Mode No. ModeType

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

plots of Fig. 6(a), showing the (velocity) ASDs from different locations of the tower. Figure 6(a) clearly shows resonant peaks at 0.59,0.71,2.46, 2.73 and 5.71 Hz; furthermore, the inspection of the spec­trai plots in Figure 6(a) clearly reveals a remarkable consistency of occurrence of spectral peaks; this infor­mation and the coherence values (Bendat & Piersol 1993), which were generally close to one in the fre­quency range where spectral peaks occur, suggest both a good quality of data and the linearity ofthe dynamic response. Specifically, it is worth noting that near unit values of coherence were generally detected in ali mea­surement points for the 1st, 2nd and 5th mode while for the other modes, especially in the lower measure­ment points, the coherence values suggest a worse signal-to-noise ratio.

As previously pointed out, in addition to the PP, the FDD procedure was used to extract the modal param­eters from ambient vibration data. The f irst singular value of the spectral matrix is shown in Fig. 6(b). The

fpp f FDD DF NMD (Hz) (Hz) (%) MAC (%)

0.586 0.598 2.05 0.9984 3.97 0.708 0. 708 0.00 0.9989 3.30 2.456 2.41 7 1.59 0.9929 8.46 2.731 2.722 0.33 0.9597 20.48 5.706 5.713 0.1 2 0.9923 8.79

results of PP and FDD methods are very similar, with the correspondence in terms of estimated resonant fre­quencies being particularly evident, since the peaks in Figs. 6(a) and 6(b) are placed practically at the same frequencies.

Table 1 summarises the identified modes and their c1assification. lt should be noticed that:

I. the classification of mode shapes is based on the availability of sensors up to the height of 48.0 m;

2. the identified frequencies are practically equal to those estimated in the 1995 field test;

3. notwithstanding the nearly symmetric shape ofthe Tower, the dominant bending modes ofthe system generally show coupled motion in the two ma in E-W and N-S directions; thus, the experimental modal analysis suggests either a strong coupling between the tower and the Cathedral or a non­symmetric stiffness distribution (as the one which has to be expected basing on the crack distribution).

444

IpI' = 0.586 Hz fi'f = 0.708 Hz fi,!' = 2.73 I Hz Jj'I, = 5.706Hz

Figure 7. Vibration modes identified during ambient vibration tests (PP).

This hypothesis was further investigated by using a 30 finite element model.

Furthermore, Table I compares the corresponding mode shapes obtained from the two different identifi­cation procedures through the frequency discrepancy DF = IUpp - !FDD)/!P?I , the MAC and the NMD. Inspection of the correlation values listed in Table I c1early highlights a very good agreement between the two methods in terms ofboth natural frequencies (with the maximum differences not exceeding 2.05%) and modal deflections (with minimum MAC value of about 0.96).

Finally, the experimentally determined bending modes ofthe tower are shown in Fig. 7.

6 STRUCTURAL MOOELING ANO UPOATING

6.1 Finite element model

A finite element model of the bell-tower was created basing on the geometric survey. The tower was mod­eled by using 8-node brick elements while the dome was represented by 4-node shell elements. A relatively large number of finite elements have been llsed in the model, so that a regular distribution of masses could be obtained and ali the main openings in the

load-bearing walls could be reasonably represented. The model consisted of 4944 nodes, 3387 solid ele­ments and 80 shell elements with 14286 active degrees offreedom. A three-dimensional view ofthe model is shown in Fig. 8.

In formulating the model , the following hypotheses were adopted:

a. the Tower footing was fixed; b. a weight per unit volume of 18.0 kN/m3 was

assumed for the masonry; c. the Poisson's ratio ofthe masonry was held constant

and equal to 0.15; d. the connection between the Southern wall of the

bell-tower and the facade of the Cathedral was accounted for by introducing rigid constraints nor­mally to the wall ; in the orthogonal direction the interaction between the tower and the Cathedral was simulated by an uniform distribution of lin­ear elastic springs of constant k (Fig. 8). The range of variation of k was estimated in order to ensure a broad correspondence between theoretical and experimental mode shapes.

A non-homogeneous distribution of the Young 's modulus was considered in order to adequately repre­sent the damaged areas of the tower; specifically, six different va lues of the elastic modulus were consid­ered in the model. Since the major cracks were placed

445

(a) (b)

Figure 8. Finile element model of lhe bell-tower: (a) 3D view; (b) West side; ( c) North-side.

N~S w

Figure 9. Distribution ofthe elastic properties up to 23.0 m.

along the East- and West- sides of the building up to about 20.0- 23.0 m, the lower part of the tower (up to 23.0 m) was divided into five sub-regions by distin­guishing the four sides and the comer properties, as it is shown in Fig. 9; a further elastic modulus was intro­duced to represent the average behavior ofthe masonry in the upper part of the tower.

Thus, the f inite element model updating was car­ried out with respect to the set of seven structural parameters summarised in Table 2:

I. the Young's moduli Ei (i = I, 2, ... ,5) in the lower part of the tower;

2. the Young's modulus E6 in the upper part of the tower;

3. the elastic constants k ofthe springs placed along the contact area between the Cathedral and the bell­tower in the direction of the Southem wall.

6.2 Finite element model updating

The uncertain structural parameters were estimated by minimizing the difference between theoretical and experimental natural frequencies. Specifically, two different well-known identification algorithms were used: the Douglas-Reid (DR) procedure (Douglas & Reid 1982) and the inverse sensitivity (/S) method (Collins et aI. 1974).

According to the DR approach (Gentile et aI. 2002), the dependence of the natural frequencies (or in gen­eral of a response parameter) of the model on the unknown structural parameters Xk (k = 1,2, . . . , N) is approximated around the current values of Xk, by the following:

(7)

where J;* represents the approximation of the i-th fre­quency of the fin ite element model and the (2N + I) coefficients Aib Bik and Ci must be determined before the evaluation ofthe unknown structural parameters by a least-square minimisation ofthe difference between eachJ;* and its experimental counterpart.f;e:

M

J = LW;C7 (8) ;=1

(9)

where Wi is a weight constant. However, eq. (7) represents a reasonable approx­

imation in a range, around a "base" value of the structural parameters xt, limited by lower Xf and upper values XF (k = 1,2, .. . , N); thus, the coeffi­cients Aik , Bik, Ci are dependent on both the base value of the structural parameters and the range in which such parameters can vary.

In the IS procedure, the natural frequencies f of the updated model are written in a Taylor series expansion as:

(10)

446

Table 2. Structural parameters of the updated models.

Structural parameter Lower value

El (N/mm2) Comer (height :::: 23 m) 800

E2 (N/mm2) West wall (height :::: 23 m) 500

EJ (N/mm2) South wall (height :::: 23 m) 800

E4 (N/mm2) East wall (height :::: 23 m) 500

Es (N/mm2) North wall (height :::: 23 m) 800

E6 (N/mm2) Ali walls (height :o: 23 m) 800

k (kN/mJ ) 34450

in which fB is the vector of the base (or prior) responses, S is the sensitivity matrix with terms aJilaXk, (X - X B) is the vector ofdifferences between updated and base parameters. Eq. (10) is then used to derive the following iteration scheme to evaluateX via I'::. X =S- ll'::.f:

(11 )

with f e being the vector of the experimental frequencies.

6.3 Comparison with the experimental results

Table 2 summarises the optimal estimates ofthe struc­tural parameters obtained from DR and IS procedures, the base value of the parameters and the assumed lower and upper limits. Furthermore, Fig. 10 shows the optimal values normalized with respect to the base value.

The inspection ofTable 2 and Fig. 10 first reveals coherent information on the stiffness distribution. The optimal estimates obtained from the two methods, although numerically different, are generally in elose agreement with the exception of the elastic modulus EJ ofthe South wall. This difference may be explained by observing that the parameter EJ slightly affects the interaction with the Cathedral facade that is mainly controlled by the spring constant k ; since the over­ali stiffness of the interaction mechanism need to be nearly unchanged, a slight decrease in the estimate of k may result in an higher increase of EJ.

The estimated parameters seem to represent quite well the damage distribution of the tower and are also in good agreement with the double flat-jack results. Specifically, a low stiffness ratio is obtained for the Eastern and Western load-bearing walls up 23.0 m, with the elastic modulus in such regions being about the half of the values obtained in the other parts of the tower. Furthermore, in the lower part of the tower the Young's modulus turns out to be higher in the cor­ner zones than elsewhere; the traditional construction,

447

XX

$ Q)

E ctJ .... ctJ a..

Updated model

Base value Upper value DR IS

1400 1400 1400 1400 1400 1400

68900

E1

E2

E3

E4 -

Es

E6

k

2500 1718 1800 930 2500 1591 1800 742 2500 1493 1800 1789

86250 69730

o Douglas-Reid, DR _ Inverse Sensitivity, IS

]

(

l íI t

~ T~

T

~ I

1772 751

2060 880

1440 1729

67987

QO DA D~ 12 1.6 Optimal Estimate I Base Value

"7

Figure 10. Optimal estimates ofthe structural parameters.

in fact , is generally characterised by a more accurate building technique in the corners.

Figure II shows the vibration modes of the DR updated model corresponding to the experimental ones and the comparison of the related natural frequen­cies. A similar correlation is obtained from IS updated model as well.

The modal parameters of the DR updated model are compared to the experimental data through the MAC and the NMD in Table 3. The natural frequen­cies ofthe DR updated model are practically equal to the corresponding experimental ones, as it has to be expected since the model updating was based on the minimisation offrequency discrepancies. The correla­tion between mode shapes shows excellent agreement with the experimental results for the first two modes (with the MAC being greater than 0.97 and the NMD less than 16.2%); for the upper modes, the correlation

i FEM ~ 0.585 Hz

i pp ~ 0.586 Hz

i FEM ~ 0.709 Hz

i pp ~ 0.708 Hz

iFEM ~ 2.455 Hz

i pp ~ 2.456 Hz

iFEM ~ 2.726 Hz

i pp ~2.731Hz

i FEM ~ 5.698 Hz

ipp ~ 5.706 Hz

Figure 11. Vibration modes ofthe updated model (Douglas-Reid method).

Table 3. Comparison between theoretical (updated model, Douglas-Reid method) and experimental modal parameters.

Mode !pp !FE:M NMD No. (Hz) (Hz) MAC (%)

I 0.586 0.585 0.9874 11.30 2 0.708 0.709 0.9745 16.19 3 2.456 2.455 0.8614 40.11 4 2.731 2.726 0.8602 40.32 5 5.706 5.698 0.8721 38.30

of mode shapes is fairly good since the MAC is always greater than 0.86 but significant average differences are highlighted by the NMD. Such differences are probably to be related ei ther to the simplified distribu­tion of the model elastic properties (which were held constant for large zones of the structure) or to a rela­tive lack of accuracy in the experimental evaluation of the higher mode shapes.

7 CONCLUSIONS

Theoretical and experimental dynamic investigation of a historic masonry bell-tower is described in the

paper. The following conclusions can be drawn from the study:

1. Within the frequency range O- 10Hz, 5 vibration modes were clearly identified;

2. The measurement of structural response to the ambient leveis of vibration mainly induced by the bells ringing has proved to be an effective means for the identification of the dynamic properties of masonry towers, although in the some measure­ment points the signal-to-noise ratio turned out to be quite low;

3. A very good agreement was found between the modal estimates obtained from classical Peak Picking (Bendat & Piersol 1993) and Frequency Domain Decomposition (Brincker et aI. 200 I) methods;

4. No experimental evidence was found that would suggest the existence ofnon-linear behavior ofthe tower during the tests;

5. The fundamental mo de of the bell-tower, 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; the coupled motion in the two main E-W and N-S directions characterizes ali the tower mode shapes;

448

6. The comparison between measured and predicted modal parameters was used to verify the assump­tions adopted in formulating the model. Specifi­cally, a good match between theoretical and experi­mental modal parameters was reached for relatively low values of the model Young modulus in the most damaged regions of the tower. Furthermore, the dynamic-based model updating, carried out by using two di fferent methods, led to consistent struc­tural parameters (distribution ofYoung's modulus in the masonry) which are in close agreement with the results of double flat-jack tests;

7. The dynamic-based assessment ofmasonry towers seems a promising approach to evaluate damage in such structures provided that an accurate geometric survey is available and in the hypothesis that the damage scenario mainly involves the lower regions ofthe building, where high eigen-sensitivity has to be expected.

ACKNOWLEDGEMENTS

The research was supported by the ltalian Ministry of Univers ity and Research (M.I.U.R.), under grant Cofin03 .

The authors are indebted with Prof. L. 8inda; with­out her involvement and advice, this research would not have been possible.

Dr. N. Gallino is gratefully acknowledged for the help in developing the application of IS identification procedure.

Furthermore, the authors would like to thank M. Antico, L. Cantin i, and M. Cucchi for the assistance in conducting the field tests.

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