# The Extreme Function Theory for Damage Detection: An Application to Civil and Aerospace Structures
## 4 Experimental Validation  
The experimental validation was performed on a well-known database realised at the Los Alamos National Laboratory (LANL). As for the frame numerical model described in the previous Section, the structure (represented in Figure 11) is a three-storey shear-type frame. Seventeen scenarios were considered, as described in Table 4. They include one or  
Figure 10: Results for the Frame FEM. (**a**) SNR = 60 dB (**b**) SNR = 40 Db Sensors channels deployed as indicated by the purple dots in Figure 5.  
more stiffness reductions applied to the beams (to simulate damage) and/or one or more added masses (to simulate changing operational conditions). The dataset also included some scenarios with a bumper-column apparatus attached at the top floor. The original intent of this mechanism was to emulate the nonlinear effects induced by a breathing crack [41], which is known to act as a pointwise source of bi-linearity (a deeper discussion can be found in Reference [42]). Generally, the presence of such nonlinearities is used to detect and localise surface cracks in otherwise linear systems [43]. However, these data are here used with another intent. As it can be noticed from Figure 12c,d, the insertion of the mechanism causes an increase in the system stiffness. Therefore, the natural frequencies grow as well, while a breathing crack would decrease them. However, the bumper-column mechanism affects the mode shapes making them detectable as a deviation from the undamaged baseline. Moreover, since the structural nonlinearities generate noise-like distortions in the frequency response, the damage scenarios 10-17 can be utilised as well to prove the robustness of the procedure when dealing with distorted signals. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Case** & **Description** \\ \hline \multirow{6}{*}{Linear States} & 1 & Undamaged Baseline \\  & 2 & Added mass of 1.2 kg at the base \\  & 3 & Added mass of 1.2 kg at the 1 st floor \\  & 4 & 87.5\% stiffness reduction in one column of the 1 st inter-storey \\  & 5 & 87.5\% stiffness reduction in two columns of the 1 st inter-storey \\  & 6 & 87.5\% stiffness reduction in one column of the 2 nd inter-storey \\  & 7 & 87.5\% stiffness reduction in two columns of the 2 nd inter-storey \\  & 8 & 87.5\% stiffness reduction in one column of the 3 rd inter-storey \\  & 9 & 87.5\% stiffness reduction in two columns of the 3 rd inter-storey \\ \hline \multirow{6}{*}{Nonlinear States} & 10 & Distance between bumper and column tip 0.20 mm \\  & 11 & Distance between bumper and column tip 0.15 mm \\  & 12 & Distance between bumper and column tip 0.13 mm \\ \cline{1-1}  & 13 & Distance between bumper and column tip 0.10 mm \\ \cline{1-1}  & 14 & Distance between bumper and column tip 0.05 mm \\ \cline{1-1}  & 15 & Bumper 0.20 mm from column tip, 1.2 kg added at the base \\ \cline{1-1}  & 16 & Bumper 0.20 mm from column tip, 1.2 kg added on the 1 st floor \\ \cline{1-1}  & 17 & Bumper 0.10 mm from column tip, 1.2 kg added on the 1 st floor \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Damage States for the Los Alamos National Laboratory (LANL) frame. Figure 11: Scheme of the Los Alamos National Laboratory (LANL) frame. Left: Side view. Right: Front view. The acquisition procedure can be summed up as follows (more details can be found in the original source [44]). Fifty band-limited (20-150 Hz) white Gaussian noise realisations were applied as input for 25.6 s at the structure base. The system response was recorded at four points corresponding to the three levels plus the base, with a sampling frequency \(f_{s}=320\) Hz. The mode shapes were then extracted from the frequency response functions (FRFs) defined between the acceleration output time histories and the force input utilising the FRVF procedure described previously. Importantly, for states 10-17, the concept of \"mode shapes\" in a nonlinear context could be not totally accurate.