# Structural Health Monitoring of 2D Plane Structures
## 2 Structural System Identification of Structures with 2D Elements by Observability Techniques
### Inverse Analysis of the Stiffness Matrix Method  
According to the Finite Element Method ([39]), the stiffness matrix of 2D element structures [K] can be obtained from the strain-displacement matrix [B] and elastic matrix [D], as presented in Equation (1):  
\[[\text{K}]=\int_{\text{V}}[\text{B}]^{\text{T}}[\text{D}][\text{B}]\text{ dV.} \tag{1}\]  
In the stiffness matrix method, matrix [K] is used to relate the equilibrium equations in terms of the nodal displacements, as presented in Equation (2). \[[\text{K}]^{(2\text{N}_{\text{N}}\times 2\text{N}_{\text{N}})}.\{\delta\}^{(2 \text{N}_{\text{N}}\times 1)}=\{\text{f}\}^{(2\text{N}_{\text{N}}\times 1)}, \tag{2}\]  
where, [K] depends on the following characteristics: thickness (h), Young's modulus (E), and Poisson ratio (\(\theta\)). \(\{\delta\}\) is the vector of displacements, which includes the vertical (\(\text{v}_{i}\)) and horizontal (\(\text{u}_{i}\)) deflections at each node i, and \(\{\text{f}\}\) is the force vector that contains the vertical (\(\text{V}_{i}\)) and horizontal (\(\text{H}_{i}\)) applied forces at each node i. Finally, \(\text{N}_{\text{N}}\) refers to the number of nodes in the finite element equation. For the case of tunnel, dam, and culvert analysis, the plane strain assumption is traditionally considered and the thickness is assumed as one. In the traditional application of the stiffness matrix method, the parameters in [K] are known and the nodal displacements can be directly obtained from Equation (2). Then, the internal forces in the elements as well as the reactions at the boundary conditions can be obtained from these displacements. This methodology is traditionally used by computer software to simulate the structural responses. A detailed explanation of the application of this approach can be found in [38]. The stiffness matrix method can be also used to identify unknown mechanical properties of the structural elements (e.g., \(\text{h}_{i}\), \(\phi_{i}\), \(\text{E}_{j}\)) from the structural response measured on site (inverse approach). In this approach, some terms in the stiffness matrix are assumed as unknown together with the reactions at the boundary conditions and the unmeasured deflections on site. In this inverse approach, non-linear products of unknown variables appear. To identify these non-linear products of unknown in 1D element structures, the OM was proposed in the literature. This method requires the rearrangement of the system of equations of the stiffness matrix method to join all the unknown variables together. To do so, the system in Equation (2) can be reordered as follows [28]:  
\[[\text{K}]^{*}.\{\delta\}^{*}=\{\text{f}\}, \tag{3}\]where, the products of variables are located in the modified vector of displacements \(\{\$^{*}\}\) and the modified stiffness matrix \([\text{K}]^{*}\) is a matrix of coefficients with different dimensions than the initial stiffness matrix \([\text{K}]\). Once the boundary conditions and the applied forces at the nodes during the nondestructive test are introduced, it can be assumed that a subset of deflections \(\delta_{1}\)* of \(\{\$^{*}\}\) and a subset of forces in nodes \(\text{f}_{1}\) of \(\{\$\}\) are known and the remaining subset \(\delta_{0}\)* of \(\{\$^{*}\}\) and \(\text{f}_{0}\) of \(\{\$\}\) are unknown. By the static condensation procedure, the system in (3) can be partitioned as follows [28]:  
\[[K]^{*}\cdot\{\$^{*}\}=\left[ \begin{array}{cc}K^{*}_{00}&K^{*}_{01}\\ K^{*}_{10}&K^{*}_{11}\end{array} \right]\cdot\left\{ \begin{array}{c}\delta^{* }_{0}\\ \delta^{*}_{1}\end{array} \right\}=\left\{ \begin{array}{c}\text{f}_{0}\\ \text{f}_{1}\end{array} \right\}=\{\text{f}\}, \tag{4}\]  
where, \(K^{*}_{00}\), \(K^{*}_{01}\), \(K^{*}_{10}\), and \(K^{*}_{11}\) are the partitioned matrices of \([\text{K}]^{*}\) and \(\delta^{*}_{0}\), \(\delta^{*}_{1}\), \(\text{f}_{0}\), and \(\text{f}_{1}\) are the partitioned vectors of \(\{\$^{*}\}\) and \(\{\text{f}\}\).