# Structural Health Monitoring of 2D Plane Structures
## 3 Example of the Application
### Statistical Analysis
With these sets, in the other words, the distributed placement of the sensors keeps the observability flow, and, consequently, FO will be obtained. If the measurements are conducted intensively at a local area or specific element of the structure, the redundancy of the measurements will emerge. In the same manner, in the case of \(\left\{2,\overline{3},\overline{5},\overline{6},8,9\right\}\) (the first set related with nine measurements), achieving the FO was due to the dispersed locations of the sensors at all nodes of the structure. Accordingly, for the same number of measurements as well as 10, the occurrence of PO in all sets (third and fifth columns of Table 5) is due to a lack of any information about node 8. For instance, choosing the measurement set, \(\left\{\overline{2},\overline{3},\overline{5},\overline{6},\overline{9}\right\}\) leads to partial observability despite 10 measurements being included in the set. This set will only allow a unique solution for the following coupled unknowns \(\left\{E_{2}.\theta_{2}.NU_{2}^{p}.NU_{2}^{N}\right\}\), \(\left\{E_{2}.NU_{2}^{p}.NU_{2}^{N}\right\}\), and \(\left\{E_{2}.NU_{2}^{p}\right\}\). However, the first set in the fourth column of Table 5, \(\left\{\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\right\}\), without any information about node 2 enabled the FO of the system. In the first recursive step of the inverse analysis, the particular solutions for the couple \(\left\{E_{1}.NU_{1}^{p}.NU_{1}^{N}\right\}\) and \(\left\{E_{1}.NU_{1}^{p}\right\}\) related to element 1 as well as \(\left\{E_{2}.\theta_{2}.NU_{2}^{p}.NU_{2}^{N}\right\}\) and \(\left\{E_{2}.NU_{2}^{p}.NU_{2}^{N}\right\}\) related to element 2 are acquired. This information is sufficient to derive the value of the parameters \(NU_{1}^{N}\) and \(\theta_{2}\) first and the rest of unknowns of the system. All the above information implies that the reason for the partial observability is that the number of effective measurements is less than the number of unknowns. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**8** & **9 Measurements** & **10 Measurements** \\ \hline P.O. & F.O. & P.O. & F.O. & P.O. \\ \hline
2, \(\overline{3},\overline{5},\overline{6},9\) & 2, \(\overline{3},\overline{5},\overline{6},8,9\) & 2, \(\overline{3},\overline{5},\overline{6},\overline{9}\) & \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) & \(\overline{2},\overline{3},\overline{5},\overline{6},\overline{9}\) \\
2, \(\overline{3},\overline{5},\overline{6},9\) & 2, \(\overline{5},\overline{6},\overline{8},\overline{9}\) & 2, \(\overline{3},\overline{5},\overline{6},\overline{9}\) & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\
2, \(\overline{3},\overline{5},\overline{6}\) & 2, \(\overline{3},\overline{5},\overline{6},8,9\) & 2, \(\overline{3},\overline{5},\overline{6},\overline{9}\) & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\  & 2, \(\overline{3},\overline{5},\overline{6},8\) & 2, \(\overline{3},\overline{5},\overline{6},\overline{9}\) & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\  & & & & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\  & & & & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\  & & & & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\  & & & & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\  & & & & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\  & & & & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\  & & & & 2, \(\overline{3},\overline{5},\overline{6},\overline{8},\overline{9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: All the sets for achieving different levels of observability for 8, 9, and 10 measurements per set. Partial observability (P.O.) and full observability (F.O.). Figure 5 illustrates some of the mentioned possible sets (in Table 5) containing both vertical and horizontal measurements.