# Structural Health Monitoring of 2D Plane Structures
## 3 Example of the Application
### Statistical Analysis  
In this part, a statistical analysis of the example is presented to indicate the required number of measurements for the partial and full observability of the system. To do so, combinatorial analysis was conducted and the following combinatory equation was used:  
\[\mathrm{C(n,r)}=(n!)/(r!(n-r)!), \tag{13}\]  
with n as the number of possible measurement and r as the chosen number of measurements in each combination. In the analyzed example, n is equal to 12, as the possible measurements correspond to \(u_{2}\), \(v_{2}\), \(u_{3}\), \(v_{3}\), \(u_{5}\), \(v_{5}\), \(u_{6}\), \(v_{6}\), \(u_{8}\), \(v_{8}\), \(u_{9}\), and \(v_{9}\). This combinatorial analysis was carried out for 8 (C\({}_{12}^{8}=495\)), 9 (C\({}_{12}^{9}=220\)), 10 (C\({}_{12}^{10}=66\)), 11 (C\({}_{12}^{11}=12\)), and 12 (C\({}_{12}^{12}=1\)) measurements per set. It can be seen in Figure 4 that the full observability of the structure requires at least nine measurements per set. In the case of eight measurements, 0.6% of the sets lead to the partial observability of the system, and 99.4% lead to an indeterminate system. For the case of nine measurements, 3.6% lead to the full observability of the system, 4.6% of the sets lead to the partial observability of the system, and 91.8% lead to an indeterminate system. This figure also shows that by conducting 10 measurements, 72.6% of the measurements lead to the full observability of the system, 1.5% of the sets lead to the partial observability of the system, and 25.9% lead to an indeterminate system. As illustrated in Table 5, to distinguish the difference between the sets achieving different levels of observability, they were classified into different patterns with regard to the location of the measurements. For instance, in the case of eight measurements \(\{\mathrm{u}|\mathrm{v}_{2},\mathrm{u}_{3},\mathrm{v}_{3},\mathrm{u}_{5}, \mathrm{v}_{5},\mathrm{u}_{6},\mathrm{v}_{6},\mathrm{u}|\mathrm{v}_{9}\}\), \(\{\mathrm{u}_{2},\mathrm{v}_{2},\mathrm{u}_{3},\mathrm{v}_{3},\mathrm{u}_{5}, \mathrm{v}_{5},\mathrm{u}|\mathrm{v}_{6},\mathrm{u}|\mathrm{v}_{9}\}\), and \(\{\mathrm{u}_{2},\mathrm{v}_{2},\mathrm{u}_{3},\mathrm{v}_{3},\mathrm{u}_{5}, \mathrm{v}_{5},\mathrm{u}_{6},\mathrm{v}_{6}\}\) were classified as \(\{2,\,\overline{3},\,\overline{5},\,\overline{6},\,9\}\), \(\{\overline{2},\,\overline{3},\,\overline{5},6,9\}\), and \(\{\overline{2},\,\overline{3},\overline{5},\,\overline{6}\}\), respectively (the vertical bar between u and v indicates that either the vertical or horizontal degree of freedom is measured). In these patterns, the subscripts of the measurements (node number) that identify the associated location are shown. Figure 4: Full observability of the system based on the various measurement sets. The node numbers with the sign (\(-\)) indicate that both deflections, vertical and horizontal, are measurements. However, no distinction is made between the measurements of X and Y degrees of freedom, as the ones without any sign are indicative of either vertical or horizontal measurements. All the sets related to the PO of the system in the cases of 8, 9, and 10 measurements are listed in columns 1, 3, and 5 of Table 5, respectively. However, columns 2 and 4 include the sets for FO of the system based on 9 and 10 measurements. One of the surprising results of the method is that, with eight targeted unknowns, at least nine measurements are required to obtain full observability. This is one of the drawbacks inherent in the method. As the nonlinear problem is being solved as a linear one and coupled unknowns are linearized, extra information might be required to uncouple them and to obtain full observability. It is clear that no set with 8, 9, or 10 measurements leads to the FO of the examined structure. Hence, if those are not elected appropriately, the end of the recursive process occurs before identifying all variables. In Table 5, the sets leading to the FO of the system are selected at dispersed locations of the structure.