IOMAC' May Guimarães - Portugal TIME-FREQUENCY ANALYSIS OF DISPERSIVE PHENOMENA IN BRIDGES

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1 IOMAC'13 5 th International Operational Modal Analysis Conference 213 May Guimarães - Portugal TIME-FREQUENCY ANALYSIS OF DISPERSIVE PHENOMENA IN BRIDGES Filippo Ubertini 1, Carmelo Gentile 2, A. Luigi Materazzi 3 ABSTRACT This paper reports on frequency splitting phenomena, also called dispersive phenomena, observed in the response of two bridges. The former is a reinforced concrete arch bridge that was tested in February 212, while the latter is an historical iron arch bridge that has been permanently monitored since November 211. The analysis of ambient vibration response data was performed through classical techniques and a more advanced time-frequency distribution. Since dispersive phenomena are typically originated by some structural damage, the presented results might contribute to the discussion towards the definition of reliable vibration-based health assessment techniques. Keywords: Frequency splitting, time-frequency analysis, bridge, operational modal analysis 1. INTRODUCTION In the literature devoted to structural health assessment there is nowadays a large consensus [1] that strategies based on (i) continuous vibration monitoring, (ii) automated modal identification [2] and (iii) multivariate statistical tools permit to identify very small localized reductions of stiffness in a structure that might be caused by structural damage. Nevertheless, the common description of damage as a localized reduction in stiffness is sometimes inappropriate to reproduce experimental observations. In a number of cases, in fact, the inspection of modal parameters estimated over a sufficiently long period of time highlights complexities in the response that cannot be simply brought back to changes in frequencies. A relevant physical phenomenon is represented by the co-existence of two close spectral peaks with similar mode shapes rather than one single mode. These phenomena are referred to as dispersive phenomena [3] and were mainly observed in the response of cracked reinforced concrete structures []. Mathematically, they can be successfully described through an additional (conservative) imaginary damping term in the case of discrete systems and through a frequency dependent elastic modulus in continuous systems. Dispersive phenomena substantially differentiate from modal hybridization because they do not consist in the interaction between two existing modes but, instead, they determine the appearance of one additional mode of vibration that does not exist in the undamaged structure. This paper reports on the observation of frequency splitting 1 Assistant Professor, University of Perugia, ubertini@unipg.it 2 Associate Professor, Politecnico di Milano, gentile@stru.polimi.it 3 Full Professor, University of Perugia, materazzi@unipg.it

2 F. Ubertini, C. Gentile, A. L. Materazzi phenomena in the response of two bridges [5-7] using classical signal processing tools and the more advanced Choi Williams time-frequency distribution [8]. 2. FREQUENCY SPLITTING AS A DISPERSIVE PHENOMENON Frequency splitting is a typical phenomenon that has been often reported in the literature as occurring in the dynamic response of cracked structural elements [3,]. It consists in the appearance of two closed frequencies in the free response of damaged systems that have very similar associated modal components and that determine the beat phenomenon. According to Zonta and Modena [3], frequency splitting of an undamped linear oscillator can be mathematically reproduced by means of the following generalized equation of motion: where is the natural circular frequency and δ is the so-called dispersion rate, representing an imaginary damping term (i is the imaginary unit). By solving the characteristic equation associated to (1) the following two natural circular frequencies can be obtained: It is important to notice that the dispersion, though appearing as a term proportional to velocity, is not associated to energy dissipation and therefore it is conservative. Again according to [3], the presence of the dissipative term responsible of the dispersive phenomenon is equivalent to considering a continuum with a Young modulus that varies linearly with the frequency, thus resulting in the following enriched one-dimensional wave equation (3), v denoting wave velocity and w being the displacement, function of space y and time t: (1) (2) (3) 3. DESCRIPTION OF THE TWO INVESTIGATED BRIDGES 3.1. R.C. road arch bridge The first bridge at study is a 7 m long reinforced concrete (R.C.) road arch bridge, owned by the Province of Perugia. The bridge crosses the Tiber river along the SR3bis secondary road, connecting the towns of Città di Castello and Umbertide in central Italy (Figure 1). The structure, located very close to the village of Montecastelli, was built as a replacement of a multi-span masonry arch bridge that was partially destroyed during the II World War. The structure of the bridge under study includes: (a) two twin arches, providing the main structural system; (b) vertical struts supporting the deck; (c) the deck structure made by a slab supported by two longitudinal members connected by transverse girders in correspondence of each vertical strut; (d) a support system consisting of a reinforced concrete side wall on fill embankment on the south side and (e) a concrete wall contrasting against the historical masonry arch bridge on the north side. The twin arches, of 68.7 m span and about 13 m rise, consist of two variable height members, with fixed width, connected by transverse girders and supporting the concrete deck through vertical struts. In the 2 m central segment, arches and deck join together. The planar distance between the twin arches is equal to the width of the masonry arch on the north side. For this reason the longitudinal girders of the deck are about 1 m external with respect to the vertical struts supported by the two arches and are resting over cantilever ends of the transverse girders. A geometrical survey and the original drawings of the bridge are not available. A few approximate geometrical characteristics are shown in the plan and elevation of Figure 1. 2

3 5 th International Operational Modal Analysis Conference, Guimarães May San Michele iron arch bridge The second case study is the historic San Michele iron arch bridge [6]. The San Michele Bridge (SMB), built in 1889 to complete one of the earliest Italian railway lines, crosses the Adda river between the small towns of Paderno and Calusco d'adda, about 5 km far from Milan, Italy. Notwithstanding the poor state of preservation of the iron structure, significantly damaged by corrosion, the historic infrastructure is still used as a combined road and railway bridge. As shown in the picture and in the elevation view of Figure 2, the SMB is characterized by the main parabolic iron arch, the upper truss box metal girder and its supporting piers. The arch spans 15 m and rises 37.5 m, thus having an aspect ratio of. It is composed by two ribs (with a height varying from 8 m at the abutments to m at the crown) that are canted inward with a planar distance of about 16 m at the basement and 5 m at the crown. The upper girder is 266 m long and it is supported by 9 equally spaced bearings, placed at a mutual distance of m. Two bearings are placed over the abutments, three are placed on the top of three piers built on masonry basements and the remaining four bearings are placed on the top of four piers supported by the main arch. The vertical trusses of the girder, 6.25 m high and 5 m apart, support two roadways: a single railroad line on the bottom and a roadway with sidewalks on the top. Figure 1 Views (top) and elevation (bottom, dimensions in m) of the investigated R.C. arch bridge Figure 2 View (top) and elevation (bottom, dimensions in m) of the San Michele Bridge 3

4 F. Ubertini, C. Gentile, A. L. Materazzi (a) (b) Figure 3 Scheme of the sensors layout in the R.C. arch bridge (a) and SMB (b) Experimental procedures An ambient vibration test (AVT) was carried out to investigate the vertical dynamic behaviour of the R.C. arch bridge at study on February 22 nd 212 [5]. Vertical acceleration responses were measured in 9 cross sections of the deck, as shown in Figure 3. In the 3 cross sections located in the middle of the bridge, sensors were placed on both sides of the deck, while in the remaining 6 sections sensors were placed only on the east side of the deck. The SMB has been permanently monitored since November 211 by the Laboratory of Vibration and Dynamic Monitoring of Structures (VIBLAB), Politecnico di Milano, and prior ambient vibration tests were also carried out since June 29 [7]. The permanent monitoring system comprises 1 vertical acceleration transducers, 7 horizontal acceleration tranduscers and two termocouples. The layout of the permanently installed sensors that measure the vertical response of the bridge is shown in Figure 3. Both in the AVT on the R.C. arch bridge and in the monitoring system of the SMB, 2-channels data acquisition systems were used (2-bit resolution, 12 db dynamic range and hardware anti-aliasing filters) with uni-axial WR 731A piezoelectric accelerometers. Each WR 731A sensor, capable to measure accelerations of.5 g with a sensitivity of 1 V/g, was connected with a short cable (1 m) to a WR P31 power unit/amplifier providing the constant current needed to power the accelerometer s internal amplifier, signal amplification and selective filtering.. TIME-FREQUENCY DATA ANALYSIS Choi-Williams (CW) time-frequency distribution was used to analyze the data recorded on the two bridges. CW distribution was chosen because it is more noise robust and more effective in reducing interference terms when compared to other time-frequency distributions (e.g. Wigner-Ville). Nevertheless, these properties are obtained at the expense of a loss in time-frequency concentration. According to Cohen s formula, time-frequency distributions can be generalized as follows: where is a kernel function, is a local autocorrelation function, is the input signal, t denotes time,. Eq. () can be rewritten in the following compact form: where is the Fourier transform of that acts as a filter of. The inspection of (5) clear shows that the choice of the kernel function essentially determines the characteristics of the time frequency distribution. The kernel of the CW distribution is defined as follows: () (5) (6) where is a suitable scalar parameter.

5 5 th International Operational Modal Analysis Conference, Guimarães May RESULTS 5.1. Analysis of the data recorded on the R.C. arch bridge Classical OMA was carried out on the R.C. arch bridge at study using several techniques [5]. A total of 8 consecutive time windows of 18 s were collected starting from 11: am. These time series, each corresponding to more than 6 times the fundamental period of the bridge, were used for the purpose of modal identification. The acceleration responses were recorded with a sampling frequency of 2 Hz. Low pass filtering and decimation were applied to the data before the use of the identification tools. More specifically, data were down-sampled to 3 Hz. Four vertical bending modes, denoted as V1, V2, etc., and five torsion modes (T1, T2, etc.) were identified from the data recorded in the eight consecutive data sets. The analysis of the results revealed that the frequency of the first antisymmetric vertical mode, mode V1, exhibited quite large variations among the different data sets. To show this, the mean values (among the eight data sets) of the natural frequencies identified in [5] are plotted together with the first singular value (SV) lines of the spectral matrix of the measurements in Figure. The results of Figure clearly outline that a single peak related to mode V1 essentially was not detected but a dispersive region, highlighted in gray, was instead evidenced. Figure First SV lines obtained from the data recorded on the R.C. arch bridge in different data sets and average natural frequencies (vertical lines) identified in [5] V1 A, Hz V1 B, Hz Figure 5 Detailed view of the dispersive region shown in Figure and mode shapes corresponding to peaks V1 A and V1 B 5

6 F. Ubertini, C. Gentile, A. L. Materazzi - V2 3 2 V1 B V1 A V2 V1 B V1 A V2 3 V Figure 6 Time frequency analyses of data recorded on the R.C. arch bridge highlighting: frequency splitting of mode V1 (a); selective excitation of lower and higher splitted modes (b); single antisymmetric vibration mode at low excitation level (c) 6

7 5 th International Operational Modal Analysis Conference, Guimarães May 213 A detailed view of the region highlighted in Figure is presented in the left plot of Figure 5. These results show that in some data sets, such as in the one recorded at 11: am and in the one recorded at 12:35 pm, two separate peaks appear for mode V1. Moreover, the frequencies corresponding to those peaks change in the two data sets. The same peaks were not detected in other cases, such as in the data recorded at 1:55 pm, where a single peak appears at a larger frequency. The mode shapes corresponding to the two peaks obtained from the data set recorded at 12:35 pm (denoted as V1 A and V1 B ) are also shown in the right plot of Figure 5. From these results it is clear that the two mode shapes were almost equivalent, which is coherent with the nature of dispersive phenomena, reviewed in Section 2. In order to better investigate the frequency variations observed for mode V1, time frequency analyses of the records collected from one single sensor were carried out. To this purpose, the transducer whose position is indicated by a circle in the mode shape plots of Figure 5 was chosen for its large associated modal component. The results of the time-frequency analysis, shown in Figure 6, pointed out that mode V1 was indeed a dispersive mode with two distinct frequency lines with large associated energy existing at the same time. It can be therefore excluded that the two peaks in the SV plots could have been originated by the variation in time of the natural frequency of one mode. The results of Figure 6 also showed that the frequency splitting of mode V1 was conceivably triggered by the vibration amplitude. Particularly, the splitting phenomenon appeared when the amplitude of vibration was sufficiently large. Moreover, the larger was the vibration amplitude the lower were the observed values of the splitted frequencies of mode V1. This last circumstance might have been a consequence of dynamic interactions with heavier vehicles passing over the bridge. Because temperature measurements were not available, it was not possible to reliably relate frequency splitting of mode V1 with this factor. However, the results of Figure showed that splitting appeared in the morning while disappeared in the early afternoon. This last circumstance might indicate that frequency splitting disappeared at larger temperatures, although it must be reported that the human feeling during the AVT reported a slightly varying temperature about equal to 6-7 C. On this respect, however, the temperature of pavement and concrete surface could have been increased during the AVT more significantly than air temperature as a consequence of solar radiation so being possibly responsible for the dispersive phenomenon Analysis of the data recorded on the San Michele Bridge As previously pointed out, a permanent dynamic monitoring system is installed in the SMB since November 211 [9]. The data are continuously recorded at 2 Hz and a binary file containing acceleration time series and temperature data, is created every hour and transmitted via Internet to Politecnico di Milano for the being processed. After extracting from each 1-hour dataset a time window of 2 s containing time series associated only to road traffic or ambient excitation, these data were low-pass filtered and decimated (after decimation, the number of samples in each record is of 96, with a sampling interval of.25 s) before applying the modal identification tools [9]. V A, Hz V B, 5.25 Hz Figure 7 First SV lines obtained from the data recorded on the SMB on December 9 th 211 7

8 F. Ubertini, C. Gentile, A. L. Materazzi The application of OMA techniques allowed to identify a total of 15 transversal bending modes and 7 vertical modes, which have been tracked through an automated modal identification procedure implemented [9] within the monitoring system. OMA results showed a frequency splitting of the fourth vertical mode (mode V). As an example, Figure 7 shows the first SV lines for the vertical acceleration data recorded on December 9 th 211. The peaks corresponding to the first four vertical modes, denoted by V1, V2, etc., are visible in this plot. At : am a clear splitting of the frequency of mode V was evidenced where the lowest peak was slightly predominant over the higher one. At 5: am mode V was still splitted but the higher frequency was prevailing. Eventually, at 3: pm the lowest peak almost disappeared and the highest peak practically became a single peak. The mode shapes associated to the two peaks detected at : am are also shown in Figure 7. They were almost identical which, again, is coherent with the dispersive model phenomenon reviewed in Section 2. In order to better investigate the splitting of mode V, Figure 8 shows the temperature recorded over the bridge on December 9 th 211 and detailed plots of the first SV lines corresponding to the data recorded at different times during that day. From these results it can be concluded that frequency splitting of mode V was conceivably triggered by temperature, with the lowest peak prevailing at low temperatures and the highest peak prevailing at high temperatures. When the temperature was sufficiently high, mode V essentially appeared as a single mode. Figure 8 December 9 th 211: Mean temperature (top) and first SV lines (bottom) obtained from the data recorded on the SMB 8

9 5 th International Operational Modal Analysis Conference, Guimarães May V B V A.5 V V.5 V Figure 9 Time-frequency analysis of selected vertical accelerations collected on the SMB on December 9 th 211: data recorded at : am (top), data recorded at 3: pm (bottom) V B V A.5 V Figure 1 Time-frequency analysis of time history of vertical acceleration recorded on the SMB on December 29 th 211 at 8: am 9

10 F. Ubertini, C. Gentile, A. L. Materazzi Frequency splitting of mode V is more clearly highlighted in the time-frequency plots of Figures 9 and 1 (data recorded on December 9 th and December 29 th 211, respectively), referring to acceleration time series recorded at the location denoted by a circle in Figure 7. The results of the time-frequency analysis of the data recorded at. am in Figure 9 clearly demonstrated the presence of two distinct frequency lines with large associated energy in the region around 5.2 Hz. A similar situation, but with the larger frequency component dominating over the lower one, was observed on December 29 th 211, Figure 1. On the contrary, on December 9 th 211 at 3: pm, bottom plot of Figure 9, mode V essentially appeared as a single mode. Amplitudes of vibration did not seem to play any role in this case. It can be concluded that mode V of SMB is also a dispersive one. 6. CONCLUSIONS Frequency splitting phenomena, consisting of the co-existence of two closed frequencies with very similar associated modal components, typically related to the presence of some structural damage, were observed and reported in the cases of two bridges: one R.C. arch bridge and one historical iron arch bridge. Because classic spectral analysis does not allow to discern between time-varying frequencies and frequency splitting phenomena, time-frequency analysis via the Choi Williams Distribution was used as a complement to classical techniques. The results demonstrated that in the case of the R.C. arch bridge the frequency splitting was triggered by the amplitude of vibration (splitting was not observed at low vibration amplitudes) while in the case of the historical iron bridge it was triggered by temperature (splitting was not observed at relatively large temperatures). The results presented in this paper can contribute to stimulate the discussion towards reliable vibration-based health-assessment techniques, highlighting that the inspection of the variations of modal parameters might not be the best possible way to detect structural damage. ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of Province of Perugia in the ambient vibration tests on the R.C. Bridge in Montecastelli (PG). REFERENCES [1] Magalhães F., Cunha A., Caetano E. (212) Vibration based structural health monitoring of an arch bridge: From automated OMA to damage detection. Mechanical Systems and Signal Processing 28: [2] Ubertini F., Gentile C., Materazzi A.L. (213) Automated modal identification and its application to bridges. Engineering Structures 6: [3] Zonta D., Modena C. (21) Observations on the appearance of dispersive phenomena in damaged structures. Journal of Sound and Vibration 21(5): [] Breccolotti M., Materazzi A.L. (2) A new damage index for the assessment of reinforced concrete structures through ambient vibration based techniques. In: Proc. 5th CETIM [5] Ubertini F., Materazzi A.L., Gentile C., Pelliccia F. (212) Auotmatic identification of modal parameters: application to a reinforced concrete arch bridge. In: Proc. EACS 212 [6] Società Nazionale delle Officine di Savigliano. Il viadotto di Paderno sull Adda (ferrovia Ponte S. Pietro-Seregno). Camilla e Bertolero, Torino; 1889 [7] Gentile C., Saisi A. (211) Ambient vibration testing and condition assessment of the Paderno iron arch bridge (1889). Construction and Building Materials 25: [8] Le K.N. (26) Time frequency distributions for crack detection in rotors A fundamental note. Journal of Sound and Vibration 29: [9] Busatta F., Gentile C., Saisi A. (212) Structural health monitoring of a centenary iron arch bridge. In: Proc. IALCCE 212: