Damage location indicator: mode shape slope ratio

Mode shapes have been widely used to figure out the damage location throughout its curvature in many researches. The node which suffers from stiffness loss has usually larger curvature than the other nodes. The mode shape is a relative quantity which can be scaled arbitrary, however, the mode shape curvature is not independent of the scaling criteria and consequently, the identification of damage location depends on how one scale the mode shape. Taking into account this fact, the concept of mode shape curvature can be modified to a quantity which is independent of scaling, and here the definition of mode shape slope ratio is introduced.

Curvature is the rate of change of the mode shape slope, to cancel out arbitrary scaling, ratio between consecutive slopes can be used instead of the rate of change of the mode shape slope. Moreover, sign of the curvature indicates the concavity. For mode shapes identified from field data, sign changes introduced in the concavity due to imperfections in the shape could make difficult the identification of damage location.

To avoid all these inconveniences, the mode shape slope ratio can be defined as follows:

(7.1)

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where mi is the slope of the i^th discrete segment of the spline interpolated mode shape.

By defining the slope ratio this way, the sign problem can be avoided, and the resultant slope ratio will only reflect how large is the slope change in a given point. At the peak point of the mode shape where the slope sign changes, the 3^rd criterion applies and it is just taking an average of the adjacent slope ratios. Finally, to avoid the a disproportionate increment in slope ratio comparing to the others, when slope in the divisor is near zero, a base 10 logarithm can be applied to the slope ratio. For the implementation, the computed mode shapes can be smoothed by curve fitting and interpolated with a spline function, which is sampled at 52 points to obtain 50 slope ratios along the bridge.

7.3.2.1 Mode shape slope ratio for test conducted in 2011/01/19

Figure 7-40 shows examples for the identified 1^st modes shape from the test in 2011/01/19 for two different time instants. Figure 7-41 shows the same but for the 2^nd mode shape.

From what is shown in Figure 7-40 and Figure 7-41, there are two identified mode shapes for both 1^st mode and 2^nd mode. The 1^st modal frequencies are separated by about 2.5 Hz one to another, and by 3 to 4 Hz for the 2^nd modal frequencies. Hence, although apparently there are three trace of frequencies revealing in the time-frequency plot shown in Figure 7-31, where the second trace is not very clear, however, both the 2^nd and 3^rd trace corresponds to the 2^nd mode. From the experience learned from section 7.1, a possible explanation of this phenomenon is the mode coupling, because the bridge is simplified to a plane model for which only horizontal vibration is measured.

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Comparing Figure 7-40 a) and b) to Figure 7-40 c); also comparing Figure 7-41 a) and b) to Figure 7-41 c) and d), there is no significant difference by visual observation, and the result with or without smoothing is similar. The 1^st mode shape slope ratio is shown in Figure 7-42 a) for the smoothed shape and b) for the shape only interpolated by a spline function.

The automatic discrimination of the mode shapes is based on the fact that, for this experimental bridge, the complex mode shape poles shown in Figure 7-40 and Figure 7-41 are almost a straight line either 1^st or the 2^nd mode, i.e., the amplitude of the poles can be treated as normal modes by only adding a plus o minus sign according to its phase. The 1^st mode has non zero crossings and the 2^nd mode has only one zero crossing.

The correlation coefficient R between the real part and imaginary part is another useful criteria to filter out spurious poles. The defined R criterion for different tests is shown in Table 7-10.

From Figure 7-42, for both a) and b), the zone with higher slope ratio become wider after 1000 seconds, indicating that the system has been changed. Initially the peak is located at the center of the bridge which is expected for a 1^st mode shape; while the scouring depth increments, the peak moves toward 300 cm, specially between 7200 and 8000 seconds as that shown in Figure 7-42 a), moments corresponding to imminent pier settlement and, precisely the pier 3 is located at 325 cm.

The 2^nd mode slope ratio is shown in Figure 7-43 a) for the smoothed shape and b) for the shape only interpolated by a spline function. Two peaks exist in the 2^nd mode shape as expected. Although the 2^nd peak is located at 325 cm (pier 3 location), but it does not change at all along the time history, otherwise, it is the first peak located at 200 cm which has a drastic movement toward 100 cm at about 7200 seconds, time instant at

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which the 2^nd modal frequency suddenly drops. Therefore, the 2^nd mode shape is not appropriate to identify the damage location in the bridge, but the drastic change of the 1^st peak position is also an indicator of imminent pier settlement.

7.3.2.2 Mode shape slope ratio for test conducted in 2011/01/26

Figure 7-44 shows examples of 1^st and 2^nd identified mode shape from different time instants. In this case the difference is evident. However, the measurement of sensor No. 12 seems to have some problem because it has consistently a phase difference with all the remaining sensors. This inconvenience is the reason of a peak slope ratio appearing at about 400 cm to 430 cm, shown in Figure 7-45 b) and Figure 7-46 b), where the mode shape is not smoothed.

From the smoothed mode shape shown in Figure 7-45 a), it is clear that the 1^st mode slope ratio is a good indicator of the damage location. In the undamaged state, the peak is low and located at the center as it is normal for 1^st mode shape; as scour occurs, the peak amplitude increases and moves rapidly to the region between 300 and 350 cm of the bridge from left to right, where is precisely the pier three location. After 7000 seconds, again, the peak moves to the region between 200 and 250 cm, where is the location of the pier two. This is because the decks on the pier three got stuck and stiffness increased, as a consequence of the fact, the peak of the slope ratio moves to pier 2 at which the scour continues.

In the other side, observing Figure 7-46 a), there are two peaks appearing in the slope ratio for the 2^nd mode which are reasonable considering nature of the 2^nd mode shape. Although a peak falls in the same region between 300 and 350 cm, there is no criterion to distinguish where the damage is located from the two peaks, besides that the

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first peak (on the top of the figure) seems to be more sensible to the scour state.

However, the same as that occurred for the 2^nd modal frequency, the 2^nd mode shape ratio can serve for the early warning purpose since position of the first peak changes suddenly just before the 1^st settlement occurs.

7.3.2.3 Mode shape slope ratio for test conducted in 2011/03/29

The mode shape slope ratio results for the test conducted in 2011/03/29 are similar to that obtained in the two previous cases. Although three modes were identified, only the 1^st mode shape slope ratio serves for the identification of damage location. Figure 7-47 shows the outcome for the 1^st mode, the 2^nd mode and the 3^rd mode are shown in Figure 7-48 and Figure 7-49 respectively.