# Structural Health Monitoring of 2D Plane Structures
## 1 Introduction
The advantages of using this deterministic and physics-based approach are: 1--the mathematical foundation of the technique is simple and comprehensible; 2--OM defines whether all the variables or a subset of them are observable or not; 3--OM permits engineers to acquire unique solutions of a structural system in a reasonable time; 4--OM allows the identification of the mechanical parameters of the structure even if the involved parameters are not linearly related; 5--if there is not a unique solution for a specific part of a system, with a quick evaluation of the system relations are obtained that allow the user to specify which measurements must be obtained to have more or complete observability [38]. OM was applied satisfactorily to solve engineering structural problems. For instance, [PERSON] et al. facilitated the application of OM for the identification of the parameters of complex concrete and steel bridges by proposing a graphical method for the selection of measurement sets, which is called observability trees [34]. The use of observability trees proved that, in the analyzed system, the probability of selecting the appropriate measurement set with a minimum number of measurements at random was practically negligible. In addition, [PERSON] et al. utilized OM to assist in decision-making and risk management processes during the maintenance and service life of structures [30]. Although it has applicability to many structures, OM has never been applied to models made out of 2D elements. Along with the undertaken studies and the proposed existing gaps, for the first time we propose a new system of equation for the application of OM to the inverse analysis of 2D models. This application is not as straightforward as it might look. The 1D OM unknowns (Young's modulus E, area A, Inertia I, and flexural or axial stiffnesses) are located only in the numerator of the stiffness matrix [28]. However, for the implementation of 2D models an obstacle arises, as the Poisson ratio appears in the both the numerator and the denominator of the elements of the stiffness matrix. This location of unknowns creates a nonlinear system of equations that prevents defining the unknowns through traditional OM. The location of the unknowns creates a non-linear system of equations that prevents the identification of the mechanical properties of the 2D element structures with the traditional OM. To solve this problem, this paper proposes a new methodology to enable the application of the OM to 2D model structures. The main contribution of this work refers to the use of changes in variables to linearize non-linear systems of equations. To illustrate the introduction of this change of variable into the observability method, a detailed mathematical analysis is presented. Finally, to validate the applicability of the method, the mechanical properties of a state-of-the-art plate are numerically determined. It is important to highlight that the proposed mathematical methodology is characterized by its generality, as it can be used for other systems of equations, including unknown variables in both the numerator and the denominator. With the proposed methodology, scholars can derive the mechanical parameters of crucial structures, such as tunnels, gravity dams, retaining walls, buttress dams, culverts, and underground pipelines. This paper is organized as follows. In Section 2, the observability technique is elucidated and we demonstrate how it could be exerted to the stiffness matrix method of 2D elements. In Section 3, an example of a state-of-the-art technique is analyzed to show the applicability of OM to 2D element structures. Moreover, a statistical analysis of the required number of measurements for reaching different levels of observability is shown in this section. Finally, the conclusions of the article are presented in Section 4.