## 1 Introduction  
Various types of sensors have been developed over the years to obtain real-time measurements of structures. Among the others, we can find those for monitoring vibration, pressure, and strain [1; 2; 3]. The most widely known strain sensors are those able to provide one-directional values in control points of the structure such as Fiber Bragg grating (FBG) sensors [4], which have been the subject of intense research and development [5] and numerous industry applications for monitoring of bridges, tanks, and concrete structures [6] since this technology overcomes some difficulties related to installation, calibration, and maintenance. A relevant example of application is the monitoring of underground pipelines. These infrastructures are aimed to transport methane gas, petroleum products, water, etc. They are usually designed for a working life of about 25-30 years. However, the lines often remain in service for a much longer time, thanks to technologies of integrity assessment. One of the main sources of risk for such infrastructures is ground movement [3; 7]. Indeed, due to the growing difficulties in finding safe routes, engineers are often forced to consider sites with a certain geological instability. In addition, in seismic areas like Italy, the country of the authors, unexpected landslides, liquefaction of sandy soils, and/or significant permanent deformations in the soil can also be caused by earthquakes. Ground movements cause additional stress, strain, and displacement states in the pipeline with respect to those produced by the operating actions such as pressure and temperature variations, which are a possible source of failure, e.g., for a stress exceeding the material strength or for deformations compromising the usability. The trend of stress and deformation states over the years is difficult to estimate reliably in the design stage. This is the reason why tools of structural monitoring are highly requested, especially for mitigating the risk in pipelines intended for fluid transport, considering that a failure may lead to serious consequences. While many studies on strain monitoring have been reported [8; 9], only a very few proposals concern the deflection monitoring of a pipeline caused by a landslide. We can cite, for instance, the proposal in [10], where the deflection curve is reconstructed by exploiting measurements coming from inclinometers. This information is an important alarm indicator, not only to prevent the failure of the line itself but also to mitigate the direct consequences of landslides on buildings and infrastructures in the affected area. The objective of this work is the development and validation of an integrated sensor-modeling tool for monitoring the displacement trend of pipelines subjected to significant ground motions. Specifically, three FBG sensors are employed at control cross-sections of the pipe. These provide three point values of the axial strain which are used for the evaluation of mean axial strain and the two bending curvatures. The hypotheses of rigid (plane) sections and negligible shear and torsional effects are assumed. Firstly, the trend of mean axial strain and bending curvatures is directly utilized to estimate the maximum normal stress over the structure. In addition, it is shown how this information can be used for predicting the displacement of underground pipelines. The procedure consists in interpolating the values of the discrete control sections in order to estimate the strain field along the pipe. Then, a 3D geometrically exact Simo-Reissner beam model [11] discretized with Lagrangian finite elements is adopted to model the pipeline. A variational formulation is developed for reconstructing the finite kinematics starting from the known strain field. A mixed iterative scheme [12] is employed for a very robust and quick solution of the nonlinear discrete equations. The model can be applied to problems with arbitrarily large displacements and rotations.