## 4 Numerical Analysis for Verification  
A tunnel structure 60 m in length was investigated to demonstrate the validity and efficiency of the proposed method. This tunnel is a reinforced concrete three-hole frame structure measuring 8.4 m in height and 33.8 m in width. The cross-section of the tunnel is shown in Figure 3a. The left hole and the right hole of the three-hole frame structure form the vehicle passage, and the middle frame is the maintenance passage. Situated 30 m underwater, the tunnel has a 2.5 m thick soil roof. The backfill soil has a floating bulk weight of 9 kN/m\({}^{3}\), a cohesion of 24.2 kPa, and an internal friction angle of 20 degrees. Additionally, the water surrounding the tunnel has a density of 996.7 kg/m\({}^{3}\).  
The load acting on the structure was calculated with a \"load-structure model\". The specific load calculation involved (1) roof surface water pressure \(\mathbf{p_{1}}=293.03\) kPa, (2) floor surface water pressure \(\mathbf{p_{2}}=375.078\) kPa, (3) covering pressure \(\mathbf{q}=-22.5\) kPa, (4) earth pressure on the top of the side wall \(\mathbf{e_{t}}=-22.855\) kPa, which is less than zero for the immersed tube section and was set as zero, and (5) the earth pressure at the bottom of the side wall \(\mathbf{e_{b}}=14.189\) kPa.  
As shown in Figure 3b, the comprehensive calculation results for the pressure on the section are as follows: \(W_{0}=p_{1}+q=315.53\) kPa, \(W_{1}=p_{2}=375.078\) kPa, \(E_{1}=p_{1}+e_{t}=293.03\) kPa, \(E_{2}=p_{2}+e_{b}=389.267\) kPa. When uneven loads are applied to the top and sides of the tunnel, the tunnel structure not only experiences vertical displacement but also lateral displacement, as illustrated in Figure 3b. Therefore, the strain generated in each section of the unit is highly complex and cannot be used directly to determine the vertical displacement of the tunnel.  
A finite element model of the tunnel structure was established in ABQUS 2020 finite element software [45], as shown in Figure 4. To eliminate the influence of boundary conditions on the numerical results, the dimensions of the model were set to 113.8 m in length, 60 m in width, and 42.5 m in depth. For the sake of simplicity, the tunnel structure was visualized as a beam to define its boundary conditions. All nodes at the bottom of the  
The left and right frames of the structure serve as channels for vehicle passage. Due to inherent challenges with sensor installation and maintenance in the two frames under normal traffic conditions, the long-gauge FBG sensors were installed in the central frame, as shown in Figure 4. Based on the sensor layout discussed in Section 3, each unit was equipped with three long-gauge sensors. A total of 180 long-gauge sensors were employed to measure the vertical displacement for 60 units. The length of the gauge is 1 m, the measurement accuracy of the sensor is 1 \(\upmu\varepsilon\), and the measurement range is between \(-600\)\(\upmu\varepsilon\) and \(3000\)\(\upmu\varepsilon\). The long-gauge FBG data were collected at a sampling rate of 100 Hz.  
Based on the coupled strain separation process discussed above, three long-gauge FBG sensors were strategically placed in the section of the pedestrian maintenance channel. This configuration effectively separated the coupled strain in the section. To test the effectiveness of the proposed coupled strain separation and tunnel vertical displacement identification method, four load distribution cases (C1-C4) were applied longitudinally along the tunnel. The \"q\" loaded on each cross-section depicted in Figure 5 represents the load distribution, which includes loads W0, W1, E1, and E2.