# The Extreme Function Theory for Damage Detection: An Application to Civil and Aerospace Structures
## 3 Numerical Simulations
### Results for the Beam (1-Dimensional) FE Model  
Figure 6 shows the results for two examples, the symmetrical pinned-pinned structural configuration and the asymmetrical clamped-pinned configuration. The method can  
Figure 5: The multi-storey FE Model. (**a**) with one column damaged at the 1 st inter-storey (\(-25\%\) or \(-50\%\)); (**b**) with one column damaged at the 2 nd inter-storey; (**c**) with one column damaged at the 3 rd inter-storey; (**d**) with both columns damaged at the 1 st inter-storey; (**e**) with two columns damaged, one at the 1 st inter-storey and one at the 2 nd inter-storey; and (**f**) with two columns damaged, one at the 1 st inter-storey and one at the 3 rd inter-storey. The purple dots indicate the position of the output channels. \begin{table}
\begin{tabular}{c c c} \hline \hline Density \(\rho\) & 2700 & kg/m\({}^{3}\) \\ Young’s modulus (columns) \(E_{column}\) & 6.90 \(\times\) 10\({}^{10}\) & Pa \\ Young’s modulus (beams) \(E_{beam}\) & 6.90 \(\times\) 10\({}^{12}\) & Pa \\ Poisson’s ratio \(v\) & 0.3 & - \\ \hline Column length (per storey) & 177 & mm \\ Beam length & 305 & mm \\ Width of the column/beam cross-section & 6 & mm \\ Height of the column/beam cross-section & 25 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Details of the frame FE model. be nevertheless applied to any statically determined or undetermined scenario. As it will be shown in the next Section, the same is valid for frame structures as well as 2-dimensional or more complex structures. Two levels of artificially added noise have been considered--SNR = 60 dB and 40 dB, respectively. The first state depicted in Figure 6 corresponds to the undamaged structure. Consider that in this state, no damage was to be detected, so the successful damage identifications and false alarms are both zeroed. The remaining states are defined for nine crack locations, equally spaced of 10% \(l\) between 0 and \(l\), and for a crack depth equal to 10% of the thickness (states 2 to 10) or 20% (states 11 to 19). It can be noticed how these states, being affected by larger damage, are more easily detected by the algorithm as foreseeable. Only the first mode shape is reported for conciseness, calculated at the nodes #10, #30, and #49 (numbered sequentially, left to right, with the beam discretised in 50 elements). Figure 6: _Cont._  
It can be appreciated that the rate of success strictly depends on the position of the damage. This issue is well-known for any mode shape-based approach. In general, this depends on the position of the mode shapes nodes and antinodes and can be solved by considering, e.g., different mode shapes at once [40]. For a baseline-based approach as the one proposed here, the normalisation to a unit maximum displacement also causes some regions of the damaged mode shape to be very close to the ones of the undamaged scenario, reducing the detectability of the damage. This arises from the eigenproblem being underdetermined and therefore the eigenvectors being defined only up to scale. This difference between the normalised damaged and baseline mode shapes is here considered in terms of cumulative Euclidean distance (CED) and it is exemplified in Figure 7 (for better interpretability, the complemental similarity 1-CED is shown). The numerical example represents the 1 st mode shape of a cantilevered aluminium box beam, modelled from the experimental case study analysed in Reference [9] (the concept can be extended to any structural configuration and set of boundary conditions and expanded to 2- and 3-dimensional structures). To make the damage effects more visible, an unrealistically large damage (a crack with deep equal to the 75% of the beam height) has been inserted at 20 equally spaced locations. The mode shapes are defined at any centimetre over the whole beam length (totalling 715 nodes). Figure 8 a,b represent the same concept adapted for the two structural configurations discussed previously.