###### Abstract  
Displacement monitoring systems play a crucial role in ensuring the safety of tunnels. Existing sensing technologies and analysis methods may be insufficient for monitoring tunnel displacement, particularly vertical displacement, due to the harshness of long-term monitoring conditions and the intricacies of structural characteristics. A long-gauge fiber Bragg grating (FBG) sensor can be used to obtain macro- and micro-level information and be connected in series for area sensing. In this study, a novel method was developed which utilizes long-gauge strain sensors to monitor the vertical displacement of a tunnel. This method employs a combination of mechanical analysis and monitoring data to accurately estimate the vertical displacement of the structure from the measured coupled strain. Several key aspects of the proposed method for identifying vertical displacement were investigated, including establishing a separation model of coupled strain on the cross-section, deriving the theory for vertical displacement identification, and determining the sensor layout of the tunnel. A series of simulation tests of a tunnel with a three-hole frame structure confirmed the efficiency and robustness of the proposed method, even when subjected to various loading conditions, noise levels, and sensor layouts. The results of this work may provide valuable insights and practical guidance for the effective and continuous displacement measurement of tunnels, ensuring their structural integrity and operational safety. vertical displacement estimation; coupled strain separation; strain monitoring; sensor layout; long-gauge FBG sensing +
Footnote †: journal: Applied sciences  
monitoring can also be conducted by combining photogrammetry and 3D scanning technology [15; 16]. Deformation and damage information can be identified simultaneously using dense 3D point cloud data [17]. Though relatively effective, these methods require the processing of massive sets of cloud data necessitating substantial computing resources and expensive equipment. It is difficult to implement them for long-term tunnel deformation monitoring. Recent advancements in strain measurement techniques have yielded remarkable results in the field of tunnel deformation monitoring [18; 19]. For example, fiber optic sensors were strategically placed along pre-tensioned gauge lengths to assess tunnel lining deformation [20]. Convergence deformation can be monitored using distributed long-gauge strain measurements, as was validated on two full-size specimens of shield segments [21]. The strain along the entire cross-section of a damaged lining can be employed to infer its deformation mode [22]. Researchers have also monitored the external and internal deformations of an inverted arch using a laser level and weak fiber Bragg (FBG) grating [23]. Robust fiber optic components and installation techniques were deployed to monitor cross-sections and assess further deformation development inside a shotcrete lining [24]. The circumferential stains in 12 rings were measured with optic fiber sensors to analyze the contribution of axial strain-to-lining deflection [25]. Yet another method for monitoring tunnel deformation was established by affixing distributed optical fiber on a PVC pipe and securing it to the tunnel's side wall [26]. A set of FBG bending meters, with angle measurement and temperature compensation functions, were employed to obtain the overall profiles of longitudinal displacement in tunnels [27]. Researchers have also combined distributed FBG monitoring technology with neural networks to develop tunnel section deformation monitoring techniques [28]. However, the majority of existing methods primarily measure the convergent deformation in tunnels and are seldom applied to accurately measure the vertical displacement of more complex structures. Advancements in fiber optic technologies have made FBG strain sensors more common in structural monitoring applications [29; 30; 31; 32].