Application 2: shaking table test of a 1-story 2-bay RC frame

A 1-story 2-bay RC frame was designed following the ACI 318-05 design code.

Dimensions and details of the RC frame can be found in Figure 7-16. The instrumentation of the RC frame can be accessed from the Figure 7-17. A total of six specimens were constructed with the same design details (denotes as RCF1, RCF2…, RCF6). This RC frame has been previously studied in [31, 32] by signal processing and nonlinear identification approaches to extract the damage feature. In [51] the RC frame is also used to trace the stiffness degradation of equivalent linear time-varying structure and the non-linear hysteretic parameters with stiffness and strength degradations.

The RC frame specimen denoted “RCF6” is used to prove the frequency tracking capability of RSSI-COV. A series of shaking table tests with increasing intensity of input excitation were applied on the specimen RCF6 to create different damage levels.

Chi-Chi earthquake ground motion data from station TCU082 (N–S direction with a

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duration of 98 s) was adopted as an input excitation to this frame. Based on the recorded maximum inter-story drift, it is shown that a different level of damaged was induced in each specimen. White noise excitation (with a low level peak amplitude defined as 30 gal) was also conducted in between each strong excitation to simulate the normal ambient vibration measurements before and after each earthquake. Block diagram shown in Figure 7-18 summarizes the excitation sequence of the test specimen on the shaking table, and the corresponding peak amplitude. A total of 97.4 s of acceleration response data were collected from the ambient vibration excitation, with a sampling rate of 200 Hz. The 800 gal* test data was not recorded.

Since this is a 1-story frame excited horizontally and in the same plane of the frame, only three accelerometers: A1, A4 and A7 are used for identification. The frame can be simplified to be a SDOF system, for which is assumed that an equivalent linear time-varying system is able to describe the lateral strength degradation. RSSI-COV is the algorithm used to trace the modal frequencies.

The model parameter of RSSI-COV is firstly chosen as follows: the moving window length is 5000 points (equivalent to 25 seconds) for more stability, number of blocks i is 200, the system order is determined to be 2 because a SDOF equivalent time-varying linear system is assumed. The statistic window length WL_statistic is 400 frequency points, displacing by every 100 points (as explained in Figure 6-1). The restart criterion is 50% of difference between this and the previous Euclidean norm of standard deviation. The time-frequency plot for the series of earthquake TCU082 scaled to different PGA is shown in Figure 7-19, recursive identification of the interspersed 30 gal white noise excitation is also shown in Figure 7-20.

In the first case, the frame is subjected to TCU082 earthquake with a PGA of 600

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gal. Before 30 seconds where the excitation is very small, RSSI-COV cannot identify correctly the undamaged state frequency, which is about 5.7 Hz as that determined by WN1. The first damage point occurs immediately after the 30 seconds, as that indicated in [32], and the identified frequency dropped to 3.9 Hz. The second point correspond to 37.66 seconds where the peak acceleration occurs. Due to the use of a moving window of 25 seconds, at this second damage the decay in frequency starts at 38.27 seconds with 3.62 Hz and ends at 40.82 seconds with 3.08 Hz. Although a larger moving window allows a more stable tracking capability, it takes more time to reflect the system change.

After the shaking of this 600 gal earthquake, the lowest frequency reached during the shaking is about 2.9 Hz, which is also consistent with that identified from 30 gal white noise excitation (WN2), which is 3.0 Hz.

The lowest frequency reached at the second test of 1000 gal TCU082 is 2.27 Hz, however, the frequency determined by WN4 is about 2.6 Hz. This may be explained by the fact that, after 1000 gal earthquake, the concrete has been severely cracked, and the lateral stiffness is highly reduced; however, at the final 10 seconds of this test where the excitation level is very small (it is about 0.02 g, i.e., 20 gal), the frequency increases to 2.4 Hz in the final segment of 1000 gal earthquake. The same phenomenon is observed for all cases: 3.6 Hz for the 1^st case with peak amplitude of 600 gal, and for the remaining cases the frequency oscillates between 2.2 to 2.5 Hz in the final segment.

These results are actually quite similar to that identified from white noise excitation (30 gal) which ranges between 2.4 and 2.6 Hz, from WN3 to WN8.

The discrepancy in the identified frequencies between strong motion and ambient vibrations can be explained by the fact that, the strength degradation of RC frame is mainly due to the concrete crack. However, the open-off of the cracks does not occur if

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the excitation level is small, i.e., the lateral stiffness is much larger when the vibration is small, therefore, higher natural frequency will be reached for the RC frame under ambient excitations.

Unlike the modal frequencies identified from the 3-story steel frame in section 7.1 which is almost totally constant, the natural frequencies identified from 30 gal white noise excitation for the RC frame shown in Figure 7-20, oscillates between 5.66 Hz and 5.78 Hz even in the undamaged state, i.e., the RC frame does not behave linearly and the signal is slightly time-varying.

From experience gathered in the simulation example of section 6, more orthogonal components than the theoretical system order is required to span the information of a time-varying system, i.e., the system order should be defined larger than 2. To avoid excessive spurious poles in the plot, a good choice would be the first 4 singular values from the singular spectrum shown in Figure 7-21, because they represent the 99.6% of total singular value powers. The frequency trace for RCF6 specimen is summarized in Figure 7-22. The moving window length is reduced to 3000 points (equivalent to 15 seconds), and consequently the trace is not as stable as it is when the window length is 5000 points, but one can expect that it can reflect the system change more quickly. The number of blocks rows i is 100.

The use of a system order of 4 has covered the segment before 30 seconds which is incorrrect when a system order of 2 is used. As that shown in Figure 7-23, signal before 24 seconds are probably a mix of measurement noise and structural response to ambient vibrations, but it were cut by the resolution of the measurements. Although the natural frequency was able to be identified from data of the first 24 seconds, this may not be reliable considering the vibration level is less than 1 gal and the resolution of the

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measurement. From Figure 7-22 a), the identified frequency corresponding to the undamaged state oscillates between 5.5 and 5.75 Hz (between 28 and 29.6 seconds), which is quite consistent with that identified from WN1. After the first damage point, the identified behavior is the same as that described previously.

Novelty Index

The Novelty Index (NI) is calculated by using the first set of the test data (from WN1 excitation) as a reference to develop the system matrix (A, C) and noise covariances (Q, R), following the procedure described in section 2.4. Based on the developed system matrix, the Kalman filter estimation was used to predict the responses for other test cases based on the procedure shown in Appendix C.

The Kalman filter prediction time history at the undamaged state (WN1) is shown in Figure 7-24. Three sensors: 1, 4, 7 are used and a system order of 8 is defined.

Although the theoretical order for a SDOF system is 2, extra components is added to cover the nonlinear and noise components, making sure that the prediction is good at the undamaged state. After 600 gal earthquake shaking, the frame was damaged and the predicted time history of WN2 is shown in Figure 7-25. The prediction error become larger than the undamaged state.

Figure 7-26 shows the change of NI for both Euclidean norm as Mahalanobis norm among different test cases. A larger index was observed for test cases subjected to severe excitation. Table 7-3 shows the results of outlier analysis (both Euclidean norm and Mahalanobis norm) from the damage detection of a 1-story 2-bay RC frame subject to a series of white noise excitations. The parameter α is set as 2, this corresponds to an interval of 95.5% confidence for a real normal distribution. The change of the identified

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dominant frequency from the test specimen with respect to the estimated NI is shown in Figure 7-27.

It is concluded that the RSSI-COV method can provide a technique for continuous monitoring of a structure by using either earthquake or ambient vibration measurement.

Although white noise excitation is assumed for the RSSI model, it is able to overcome this assumption violation by increasing the subspace dimension (number of block rows) which allows a better signal decomposition, but a higher system order is required to cover the system information. Drawback is the spurious poles that appear in the time-frequency plot due to the assumption violation.

On the another side, the novelty index analysis based on the Kalman filter prediction error can provide the possibility of getting early warning of structural damage before severe damage occurs by using output-only measurements.