## 3 Displacement Estimation Method
### Coupled Strain Separation  
During the service life of the tunnel, a long-gauge FBG sensor captures strain data that encompass three-directional components (vertical and transverse bending strain, axial strain). To accurately determine the vertical displacement, it is necessary to separate the vertical bending strain from the coupled strain. According to material mechanics, the normal stress at any point of a section in a linearly elastic material is proportional to the distance between the point and neutral axis. By selecting three points arranged in an isosceles triangle on the section as monitoring points, it becomes possible to decouple the coupled strain. Three monitoring points are essential for determining the three unknown strains. Isosceles triangles with simple geometric relationships can be utilized to simplify the many complex calculations involved in this process. The relationship between the coupled and separated strain in the form of an isosceles triangle is depicted in Figure 2.  
In Figure 2, \(y\) and \(z\) are the neutral axes of vertical bending strain and transverse bending strain, respectively. At any given point in the section, the collected strain is the sum of the strains in all three directions. \[ \begin{cases}\varepsilon_{1}=\varepsilon_{N1}+\varepsilon_{my1}+\varepsilon_{ mz1}\\ \varepsilon_{2}=\varepsilon_{N2}+\varepsilon_{my2}+\varepsilon_{mz2}\\ \varepsilon_{3}=\varepsilon_{N3}+\varepsilon_{my3}+\varepsilon_{mz3}\end{cases}  \tag{2}\]  
where \(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\) are the monitored strains at points 1, 2, and 3; \(\varepsilon_{N}\), \(\varepsilon_{my}\), \(\varepsilon_{mz}\) represent axial, vertical bending, and transverse bending strains, respectively. Axial strains on the same section are considered to be equivalent, that is, \(\varepsilon_{N}^{\prime}=\varepsilon_{N1}=\varepsilon_{N2}=\varepsilon_{N3}\), which can be substituted into Equation (2) to derive Equation (3):  
\[ \begin{cases}\varepsilon_{1}=\varepsilon_{N}^{\prime}+\varepsilon_{my1}+ \varepsilon_{mz1}\\ \varepsilon_{2}=\varepsilon_{N}^{\prime}+\varepsilon_{my2}+\varepsilon_{mz2} \\ \varepsilon_{3}=\varepsilon_{N}^{\prime}+\varepsilon_{my3}+\varepsilon_{mz3} \end{cases}  \tag{3}\]  
Based on elastic theory, the vertical bending strain at point 1 can be expressed as  
\[\varepsilon_{my1}=\frac{Myz_{1}}{EI_{y}}, \tag{4}\]  
Figure 2: Relationship between three points: (**a**) monitoring point position; (**b**) vertical bending strain; (**c**) transverse bending strain. where \(M_{y}\) is the vertical bending moment; E is the elastic modulus; \(I_{y}\) is the moment of inertia with respect to the y axis; \(z_{1}\) is the distance from point 1 to the neutral axis \(y\). In Figure 2a, \(z_{2}=z_{3}\), so the vertical bending strains at points 2 and 3 are:  
\[ \begin{cases}\varepsilon_{my2}=\frac{M_{y}z_{22}}{EI_{y}}=\frac{M_{y}(z_{1}-h)} {EI_{y}}\\ \varepsilon_{my3}=\frac{M_{y}z_{3}}{EI_{y}}=\frac{M_{y}(z_{1}-h)}{EI_{y}}\\ \end{cases}  \tag{5}\]  
where \(z_{2}\) and \(z_{3}\) are the distance from points 2 and 3 to the neutral axis \(y\), respectively; \(h\) is the vertical distance from point 1 to points 2 or 3. The vertical bending strain at points 2 and 3 can be directly correlated to the vertical bending strain at point 1 by combining Equations (4) and (5):  
\[\varepsilon_{my2}=\varepsilon_{my3}=\frac{z_{1}-h}{z_{1}}\varepsilon_{my1}. \tag{6}\]  
Similarly, the transverse bending strains at points 1, 2, and 3 are:  
\[\varepsilon_{m21}=\frac{M_{z}y_{1}}{EI_{z}},\;\varepsilon_{m22}=\frac{M_{z}y_{ 2}}{EI_{z}},\;\varepsilon_{m23}=\frac{M_{z}y_{3}}{EI_{z}}, \tag{7}\]  
where \(M_{z}\) is the transverse bending moment; \(I_{z}\) is the moment of inertia with respect to the z axis; \(y_{1},y_{2},y_{3}\) are the distances from points 1, 2, and 3 to the neutral axis \(z\).