PSI - Issue 64

Marco Pirrò et al. / Procedia Structural Integrity 64 (2024) 669–676 Author name / Structural Integrity Procedia 00 (2019) 000–000

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A well-assessed minimization procedure is based on the PCA algorithm (Sharma, 1995), which is a multi-variate technique that finds the variability common to the identified natural frequencies. Once trained in a period sufficiently long to cover a wide range of EOV, the PCA regression model is used to reconstruct the natural frequencies and the residuals between the identified and predicted ones should contain only the effects of possible structural variations. In the following, the PCA residuals are used to construct a multi-variate T 2 -Hotelling control chart (Hotelling, 1947) with the Upper Control Limit (UCL) set equal to the 95th percentile of the residuals calculated during the training period. In the T 2 -Hotelling control chart, groups of 96 residuals are selected, thus one T 2 -value is computed every 2 days. Alternatively, to PCA + T 2 -Hotelling control chart, the Cointegration procedure (Cross et al. 2011) is being adopted for structural assessment purposes only in recent years. Given M non-stationary time-series Y k Î Â N ´ M (e.g. N observations of M natural frequencies), they are said to be cointegrated if a stationary linear combination e k Î Â N of Y k exists (Eq. 1):

(1)

The unknown coefficients b j ( j = 1, …, M ) in Eq. (1) can be found using the Johansen procedure (Johansen, 1988) during a training period in which the structure is supposed to be in normal condition under typical EOVs. Johansen procedure evaluates a Vector Error Correction Model (VECM) of order p from Y k , as:

(2)

where the matrices PÎÂ N ´ M and B j ÎÂ N ´ M describe, respectively, the long-run equilibrium and short-run adjustment of the time-series Y k in order to maintain them in equilibrium. A very-refined maximum-likelihood estimation aims to find the coefficients b j (Eq. 1) that are contained in the first row of matrix P (Eq. 2). Once found the unknown coefficients, the resulting linear combination e k is a new time series – called cointegration residual – purged from the common trends of the natural frequencies and mainly due to EOVs. Hence, the anomaly occurrence is detected when the cointegration residual deviates from its stationarity highlighted in the training period. In the following, the upper and lower control limits (UCL, LCL) are set as the 95th percentile of the cointegration residual e k computed in the training period. Structural anomalies are detected each time an observation consistently lies outside the control limits: therefore, the cointegration residual is directly interpreted as the damage sensitive feature, as well as the T 2 -Hotelling statistic built on the PCA residuals. The present section compares the results obtained from the PCA + T 2 -Hotelling control chart approach and the Cointegration procedure. For comparison purposes, three different choices of training period lengths are selected: a) 3 months (i.e., from 01/12/2015 to 29/02/2016), b) 6 months (i.e., from 01/12/2015 to 31/05/2016) and c) 12 months (i.e., from 01/12/2015 to 30/11/2016). From Fig. 6 it’s possible to conclude that: • Increasing the training period (Fig. 6c), the two approaches are able to account for the environmental effects (mainly temperature and water level), since the T 2 -Hotelling statistic and the cointegration residual maintain their stationarity after the training period. Furthermore, the occurrence of the outliers is negligible, proving the absence of any structural anomaly during the monitoring period (Fig. 6c); • Comparing the two approaches, it can be noted that a training period of 6 months is enough for Cointegration procedure. Instead, the PCA + T 2 -Hotelling control chart approach needs at least 12 months of training to be able to correctly minimize the EOVs. Consequently, the presented results clearly highlight the efficiency of Cointegration procedure for the structural assessment of the dam, since it requires only 6 months of training period to construct a linear combination of the