# Structural Health Monitoring of 2D Plane Structures
## 3 Example of the Application
### Inverse Analysis  
In this section, the proposed algorithm is used for the inverse analysis of the structure from the following measurement set (u\({}_{2}\), v\({}_{2}\), u\({}_{3}\), v\({}_{3}\), u\({}_{5}\), v\({}_{5}\), u\({}_{6}\), v\({}_{6}\), u\({}_{9}\), and v\({}_{9}\)). The numerical values of these deflections correspond with those presented in Table 4.  
To identify the observable parameters in the structure, the stiffness matrix system described in Step 1 of the algorithm was calculated. This example contains 18 components for displacements (9 for both u and v). Unfortunately, the large size of this matrix (18 \(\times\) 18) prevents its representation in the paper and only the first equation (the one related to the horizontal force at node 1, H\({}_{1}\) is depicted) is presented in Equation (10):  
\[H_{1}=\left[ \begin{array}{c}((E_{1}.h_{1})/2(\vartheta_{1}+1))+((E_{2}.h_{2} )/(\vartheta_{2}-1))/((4.(2\vartheta_{2}-1).(\vartheta_{2}+1))\\ 0\\ ((E_{2}.h_{2}).(\vartheta_{2}-1))/(3.(2\vartheta_{2}-1).(\vartheta_{2}+1))\\ -(2.E_{2}.h_{2}.(\vartheta_{2}-1))/(3.(2\vartheta_{2}-1).(\vartheta_{2}+1))\\ (E_{2}.h_{2}.(\vartheta_{2}-1))/(12(2\vartheta_{2}-1).(\vartheta_{2}+1)\\ (E_{2}.h_{2}.\vartheta_{2})/6.(2\vartheta_{2}-1).(\vartheta_{2}+1)\\ -(2.E_{1}.h_{1})/3.(\vartheta_{1}+1)\\ (E_{1}.h_{1})/3.(\vartheta_{1}+1)\\ 0\\ ((2.E_{2}.h_{2}.\vartheta_{2})/(3(2\vartheta_{2}-1).(\vartheta_{2}+1)))-((E_{1 }.h_{1})/3(\vartheta_{1}+1))\\ 0\\ 0\\ 0\\ (E_{1}.h_{1})/(6(\vartheta_{1}+1))\\ -(E_{1}.h_{1})/(12(\vartheta_{1}+1))\\ 0\\ 0\\ 0\\ ((E_{1}.h_{1})/2(\vartheta_{1}+1))-((E_{2}.h_{2}.\vartheta_{2})/(6(2\vartheta _{2}-1).(\vartheta_{2}+1))\\ \end{array} \right]^{T}\left[ \begin{array}{c}u_{1}\\ v_{1}\\ u_{2}\\ v_{2}\\ v_{3}\\ u_{4}\\ v_{4}\\ u_{5}\\ v_{5}\\ u_{6}\\ v_{7}\\ u_{8}\\ v_{8}\\ u_{9}\\ \end{array} \right] \tag{10}\]  
As shown in Equation (10), Poisson's ratio appears in both the numerator and denominator of the matrix parameters. This location of Poisson's ratio prevents the application of traditional OM (proposed in the literature) and makes necessary the changes in variables proposed in Step 2. After this change in variables, the system of equations can be rearranged. In this way, the size of the modified stiffness matrix is increased to 18 \(\times\) 42. Again, for size limitations only the obtained equilibrium equation of H\({}_{1}\) is presented in the paper. This equation is as follows:  
\[H_{2}=\left[ \begin{array}{c}0\\ 0\\ 0\\ 2\times 10^{-07}\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{array} \right]^{T}\left\{ \begin{array}{c}E_{1}.\theta_{1}.NUl_{1}^{P}.NUl_{1} ^{N}\\ E_{1}.\theta_{1}.NUl_{1}^{P}.NUl_{1}^{N}.u_{8}\\ E_{1}.\theta_{1}.NUl_{1}^{P}.NUl_{1}^{N}.v_{8}\\ E_{2}.\theta_{2}.NUl_{2}^{P}.NUl_{2}^{N}\\ E_{1}.NUl_{1}^{P}.NUl_{1}^{N}\\ E_{1}.NUl_{1}^{P}.NUl_{1}^{N}.u_{8}\\ E_{1}.NUl_{1}^{P}.NUl_{1}^{N}.v_{8}\\ E_{1}.NUl_{1}^{P}.v_{8}\\ E_{2}.NUl_{2}^{P}\\ E_{2}.NUl_{2}^{P}.NUl_{2}^{N}\\ H_{1}\\ H_{2}\\ V_{1}\\ V_{4}\\ V_{7}\end{array} \right]. \tag{12}\]  
This equation shows that at this stage, the number of unknown variables in the system is eight (E\({}_{1}\), \(\phi_{1}\), \(\rm{NU_{1}^{P}}\), \(\rm{NU_{1}^{N}}\), E\({}_{2}\), \(\phi_{2}\), \(\rm{NU_{2}^{P}}\), \(\rm{NU_{2}^{N}}\)). To identify the observability of the variables, the null space [V] was numerically calculated in Step 7. With this information, the general solution of the system {z} described in Equation (6) can be written as presented in Figure 3.  
The analysis of the null space presented in Figure 3 shows that the partial observability of the structure is obtained. The particular solution of these parameters was numerically obtained with the pseudoinverse of matrix [C] in MatLab (Step 8). In Figure 3, the observable parameters (the ones with a unique solution) are associated with the null rows of the matrix [V].