## 8 Case Study Application  
The cantilever steel column considered before is analyzed with some modifications to test the RUL calculation. First, the damage of the column (i.e., the corrosion thickness) was correlated with time, assuming a continuous degradation rate equal to 80 micrometers/year, which is the reduction associated with an urban-industrial environment [49]. Figure 15 shows the framework developed and implemented. Herein, the gray-highlighted portion allows creating benchmark data to test the RUL algorithm. The whole framework is repeated whenever the damage evaluation is required, thus allowing the update of the parameters of the degradation model. As shown in Figure 15, the approach is very similar to the AI training described in the previous section. The lumped mass at the top of the column was introduced in stochastic terms, the signal at the base was reassigned in each iteration as a pseudo-random white-noise signal to simulate environmental vibrations, and the damage parameterization was modified to account for a time-dependent damage function (in this case a function governing the corrosion rate). The following parameters have been used:  
1. corrosion rate: 80 micrometers/year
2. simulated operating period: 60 years
3. number of damage estimations for each simulated year (i.e., number of runs of the trained AI to estimate the corroded thickness): 10  
Figure 15: Framework for estimating RUL. input signal: pseudo-random white noise signal, with a length of 10 min and a sampling frequency of 200 Hz, for each damage identification run
5. safety threshold: 5 mm of corroded thickness  
Figure 16 shows the comparison between the thickness lost in the corrosion process associated with the assumed corrosion rate, i.e., the thickness reduction entered in the FE model, as well as the AI prediction. The discontinuity of the latter is due to the AI training phase, where the recognition of a corrosion thickness with a sensitivity of 1 mm was set. One could have opted to create a training scheme with higher sensitivity, say 0.5 mm, in the training phase. However, this would certainly have lowered the accuracy, as the AI would have struggled more to recognize a smaller variation in the extracted features (e.g., natural frequency). The number of analyses to be performed is another important aspect affecting the choice of the training scheme. As mentioned before, seeking a more sensitive and accurate AI results in an increase in the number of signals to be created, which may be far from negligible. Herein, the proposed AI represents a good compromise between training time, sensitivity, and accuracy, even if it sets up a discontinuous trend. The best compromise between computational time and sensitivity will be the object of future developments. Figure 17 shows the output of the RUL algorithm (Figure 15). The algorithm is run for each new identification, and at each time point, the RUL is recalculated by updating the parameters. The diagram above shows the comparison between the real RUL (RULreal), i.e., the remaining useful life calculated by knowing the confidence limit (5 mm of corrosion) and the corrosion rate function, as well as the RUL obtained by updating the degradation model (RULtest). The confidence interval of the latter is also given. The bottom graph shows the ratio RULtest/RULreal in percent. One important aspect immediately stands out: the estimated RUL almost overlaps with the actual RUL in the period between about 15 and 40 years. This proves the excellent estimation of the forecast.