# Structural Health Monitoring of 2D Plane Structures
## 1 Introduction  
In recent years, the maintenance, health monitoring, and identification of structural systems are becoming more frequent all around the world [1, 2]. A methodology to identify the mechanical properties of existing structures according to in situ measurements is defined as structural system identification [3] and is a necessary part of any Structural Health Monitoring system. In the maintenance and rehabilitation process of the structures, in addition to the in situ measurements and visual inspections, precise damage detection might be performed using numerical and analytical analysis [4, 5]. Examples of non-model-based damage identification approaches can be found in [6, 7]. Many structural system identification methods have been proposed for estimating the mechanical parameters of structures modelled with 1D elements, such as steel and concrete buildings as well as cable-stayed bridges, trusses, and frames [8, 9, 10]. However, despite the intricate nature of 2D structural models such as tunnels, dams, and culverts, few investigations have been conducted on the structural system identification of these structures [11, 12, 13, 14, 15]. For instance, geotechnical engineers who tried to estimate the behavior of buried structures encountered insufficient sets of input data [16, 17]. In the case of segmental underground structures, both the constraining effect of the soil and the existence of the segment joints will generate problems for acquiring dynamic characteristics [18, 19, 20]. [PERSON] et al. presented a new intelligent inverse analysis technique combining fuzzy systems, an imperialistic competitive algorithm, and numerical analysis for the back analysis of the Karaj Subway in Iran [21]. [PERSON] et al. determined the geotechnical parameters using inverse analysis based on convergence data [22]. They also proved that their proposed method would be a more economical and time-saving technique in comparison to a design based on soil mechanic tests carried out by consultant engineers. Structural system identification methods can be categorized as parametric and non-parametric [23, 24, 25]. With the quick increase in computer technology, jutting numerical software, and well-known packages such as MatLab or Mathematica, the popularity ofnon-parametric methods has increased. These methods define the transfer functions of a system, which implies that the input-output relation is characterized by a set of equations that may not have any explicit physical meaning. [PERSON] et al. estimated the potential use of the differential evolution genetic algorithm in the back analysis of a tunnel response to obtain improved estimates of the model parameters by matching the model prediction with the monitored response [25]. [PERSON] et al. proposed two algorithms by means of automatically generating optimal measurements [26]. They proved the validity of their methods with academic examples. They used their proposed method for the Munich subway tunnel project. [PERSON] et al. presented a procedure to carry out back analysis of the tunnel excavation problem in an automatic manner [27]. They verified their measurements with ones obtained from a finite element analysis. However, in parametric methods all the parameters have physical meaning in the equations and may be used as a method to identify the parameters of the structures. [PERSON] et al. proposed a parametric method to uniquely identify the mechanical properties of structures as well as the flexural and axial stiffness (EA, EI) [28]. This technique is called the observability method (OM) and is enforceable for the structural system identification of 1D elements (Bernoulli and Timoshenko beam elements) but not for finite element models with 2D elements. OM stands as one of the unique structural system identification parametric approaches presented in the literature for the static [29; 30; 31; 32; 33; 34; 35; 36] and dynamic analysis of the stiffness matrix method [37].