Vertical Displacement Measurement of Tunnel Structures Based on Long-Gauge Fiber Bragg Grating Strain Sensing

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.

1. Introduction

Tunnel deformation is not only a major phenomenon that affects the safety of a tunnel under service, but it is also responsible for ambient stress changes that impact the safety of tunnels [1]. Prolonged uneven settlement of the structure is one of the most problematic aspects of tunnel operation, which primarily manifests as vertical displacement in the structure [2,3,4,5]. Therefore, it is crucial to continuously monitor the vertical displacement of the tunnel to ensure its safety. Traditional deformation monitoring techniques involve the regular use of equipment such as levels, total stations, and cable displacement meters to measure specific target areas at fixed points [6,7,8,9]. However, these techniques only measure one or a few points and require fixed references.

To mitigate these problems, many researchers have proposed tunnel displacement identification methods based on point clouds [10]. Three-dimensional (3D) matching algorithms, for example, can be employed to evaluate wall displacement based on the specific conditions of tunnel surfaces [11]. A 3D modeling algorithm was used to process the raw point cloud for displaying tunnel deformation in a previous study [12]. Burdensome metadata processing and low-accuracy data analysis issues can be addressed systematically [1]. The overall and local deformation of a tunnel can be determined via denoising and 360° deformation analysis of point cloud data [13]. Profiles from mobile tunnel laser scanning systems can be utilized to observe the cross-sections of tunnels [14]. Tunnel deformation 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]. Fiber optic sensors can be classified into point, distributed, or long-gauge types. The FBG point sensor can measure small-sized areas, providing potentially highly precise measurement values. When utilizing a commercialized acquisition instrument, the FBG point sensor can achieve accuracy of less than 1 με within a strain range of 2500–5000 με [33]. However, it is important to note that the accuracy of point measurements can be influenced by the presence of discontinuities. Measurement points must be carefully selected to ensure accurate results [34].

Distributed sensing technologies offer extended measurement distances based on Rayleigh and Brillouin scattering, allowing for absolute measurements rather than reflecting merely relative changes compared to reference measurements [35]. In comparison to conventional Brillouin optical time domain analysis (BOTDA), which has a spatial resolution limited to approximately 0.5 m, improved BOTDA techniques can achieve spatial resolution of up to 20 mm [36]. However, the strain resolution typically remains low, often exceeding 20 με [21,37]. Long-gauge fiber optic sensors also enable measurement of the average strain between two separate points, unlike conventional sensors, making them suited to engineering structures with heterogeneous materials such as concrete [38]. Long-gauge FBG sensors allow for the precise measurement of average strain over a significant gauge length, spanning several centimeters or meters. This feature enables the high precision measurement of various parts of a structure or the entire structure itself, particularly for displacement measurement (with an error rate of less than 5%). Such sensors have been effectively employed in numerous structure monitoring applications [39,40].

In this study, long-gauge FBG sensing was employed to identify the vertical displacement of a tunnel. The strain of the tunnel section was analyzed using a mechanical model for the establishment of the coupled strain separation model. The vertical displacement of the tunnel was then calculated by the proposed method.

The remainder of this paper is organized as follows. Section 2 introduces the sensing principle of the long-gauge FBG sensor, and Section 3 describes the proposed method for measuring vertical displacement and its theoretical basis. Section 4 discusses the numerical modeling of the investigated tunnel. Section 5 provides a brief summary and concluding remarks.

2. Long-Gauge FBG Sensing Principle

A long-gauge FBG sensor was developed to measure the average strain over extended gauge lengths [41,42]. This sensor can be customized for specific lengths, spanning from several centimeters to meters, making it ideal for the on-site monitoring of large civil structures. Installing a long-gauge FBG sensor involves the insertion of a bare fiber with FBG into an embedded tube and fixing both ends to ensure that the measured value represents the average strain over the measured length. The bare optical fiber, including the FBG, serves as the sensing component and operates effectively, apart from temperature compensation. This compensation issue can be resolved by incorporating a dummy-reference Bragg grating subjected to the same thermal environment but free from any load. Temperature compensation for the long-gauge strain sensor can be achieved by subtracting the response of the FBG temperature sensor from the response measured by the strain sensor.

The internal structure of the long-gauge FBG sensor is shown in Figure 1a. The long-gauge FBG sensor outputs static and dynamic structural strains related to the structural angle. For a beam element with two local degrees of freedom per node (vertical displacement w and angle θ), the long-gauge strain (macro strain) of the sensing element can be expressed as follows:

$${\epsilon }_j(t)={\mathsf{\mu }}_j[{\theta }_o\left(t\right)-{\theta }_p\left(t\right)]$$

(1)

where ${\mathsf{\mu }}_j={\mathrm{h}}_j/L_j$, j represents the jth sensor unit; h is the distance from the sensor to the neutral axis of the beam section; L is the length of the sensing unit; ${\theta }_o\left(t\right)$ and ${\theta }_p\left(t\right)$ are the angular degrees of freedom of o and $p$ at both ends of the sensor unit at time t, respectively.

The accuracy of strain measurement is consistently maintained at 1–2 με when utilizing commercial demodulators. Additionally, this device can achieve a dynamic measurement frequency of up to 1000 Hz. The long-gauge FBG strain sensor examined in this study was initially calibrated in the laboratory to determine the calibration coefficient. The measured responses from sensors could then be calibrated using this coefficient. Multiple long-gauge FBG sensing units were connected in series to cover a critical sensing area. These key areas can then be interconnected to form a distributed sensor network for macro-strain measurement in structural regions. Figure 1b illustrates the area sensing capabilities of the long-gauge FBG sensor.

3. Displacement Estimation Method

External factors (e.g., earth pressure, water pressure, vehicle loads) create strain throughout sections of a tunnel during service. The strain measured at any point within the tunnel provides crucial information regarding the vertical, transverse, and axial displacement of any given section. A two-step approach was used in this study to accurately assess vertical displacement. First, the monitored coupled strain was separated to obtain the vertical bending strain. Second, the vertical bending strain separated from the coupled strain was used to estimate the vertical displacement of the structure.

3.1. 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.

$$\left\{\begin{array}{c}{\epsilon }_1={\epsilon }_{N1}+{\epsilon }_{my1}+{\epsilon }_{mz1}\\ {\epsilon }_2={\epsilon }_{N2}+{\epsilon }_{my2}+{\epsilon }_{mz2}\\ {\epsilon }_3={\epsilon }_{N3}+{\epsilon }_{my3}+{\epsilon }_{mz3}\end{array}\right.$$

(2)

where ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$ are the monitored strains at points 1, 2, and 3; ${\epsilon }_N$, ${\epsilon }_{my}$, ${\epsilon }_{mz}$ represent axial, vertical bending, and transverse bending strains, respectively. Axial strains on the same section are considered to be equivalent, that is, ${\epsilon \prime }_N={\epsilon }_{N1}={\epsilon }_{N2}={\epsilon }_{N3}$, which can be substituted into Equation (2) to derive Equation (3):

$$\left\{\begin{array}{c}{\epsilon }_1={\epsilon \prime }_N+{\epsilon }_{my1}+{\epsilon }_{mz1}\\ {\epsilon }_2={\epsilon \prime }_N+{\epsilon }_{my2}+{\epsilon }_{mz2}\\ {\epsilon }_3={\epsilon \prime }_N+{\epsilon }_{my3}+{\epsilon }_{mz3}\end{array}\right.$$

(3)

Based on elastic theory, the vertical bending strain at point 1 can be expressed as

$${\epsilon }_{my1}=\frac{{M_{y}z}_1}{E{I}_y},$$

(4)

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:

$$\left\{\begin{array}{c}{\epsilon }_{my2}=\frac{M_y{z}_2}{E{I}_y}=\frac{M_y(z_1-h)}{E{I}_y}\\ {\epsilon }_{my3}=\frac{M_y{z}_3}{E{I}_y}=\frac{M_y(z_1-h)}{E{I}_y}\end{array}\right.$$

(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):

$${\epsilon }_{my2}={\epsilon }_{my3}=\frac{z_1-h}{z_1}{\epsilon }_{my1}.$$

(6)

Similarly, the transverse bending strains at points 1, 2, and 3 are:

$${\epsilon }_{mz1}=\frac{{M_{z}y}_1}{E{I}_z}, {\epsilon }_{mz2}=\frac{{M_{z}y}_2}{E{I}_z}, {\epsilon }_{mz3}=\frac{{M_{z}y}_3}{E{I}_z},$$

(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 $z$. Based on the geometric relationship between points 2 and 3, the transverse bending strain at point 3 can be expressed in terms of the corresponding strain at point 2:

$${\epsilon }_{mz3}=\frac{y_2-d_3}{y_2}{\epsilon }_{mz2},$$

(8)

where $d_3$ represents the distance between points 2 and 3. The relationship between points 2 and 3 is shown in Figure 2c. As indicated in Figure 2a, $y_2=\frac{d_3}{2}$, which can be substituted into Equation (8) as follows:

$${\epsilon }_{mz3}=-{\epsilon }_{mz2}$$

(9)

Because point 1 lies on the neutral axis z, $y_1=0$, the transverse bending strain at point 1 is

$${\epsilon }_{mz1}=0.$$

(10)

Finally, Equations (6), (8)–(10) can be substituted into Equation (3) to establish the following:

$$\left\{\begin{array}{c}{\epsilon }_1={\epsilon \prime }_N+{\epsilon }_{my1}\\ {\epsilon }_2={\epsilon \prime }_N+\frac{z_1-h}{z_1}{\epsilon }_{my1}+{\epsilon }_{mz2}\\ {\epsilon }_3={\epsilon \prime }_N+\frac{z_1-h}{z_1}{\epsilon }_{my1}-{\epsilon }_{mz2}\end{array}\right.$$

(11)

Solving Equation (11) yields ${\epsilon \prime }_N$, ${\epsilon }_{my1}$, ${\epsilon }_{mz2}$. The vertical bending strain and transverse bending strain can be calculated at the three points using Equations (6) and (10). This enables the coupled strain monitored at the three points to be decomposed into three distinct directional strains.

3.2. Vertical Displacement Estimation

Stress characteristics were examined using a vertical bending strain that had been separated from the coupled strain, as discussed in this section. Considering the distribution characteristics of the vertical bending deformation of a tunnel, it is appropriate to model the tunnel as a semi-infinite elastic foundation beam. A conjugate beam method (CBM) was applied to accurately calculate the vertical displacement of the tunnel using the separated strain [43,44].

The conjugate beam theory takes the bending moment distribution M(x) on the actual beam as the load distribution on the conjugate beam, then the corresponding bending moment distribution on the conjugate beam is equivalent to the vertical deformation distribution y(x) of the actual beam. For a Euler beam, the relationship between its bending strain, curvature, and bending moment can be expressed as follows:

$$\mathrm{k}\left(\mathrm{x}\right)=\frac{{\mathsf{\epsilon }}_{\mathrm{m}\mathrm{y}1}(\mathrm{x})-{\mathsf{\epsilon }}_{\mathrm{m}\mathrm{y}2}(\mathrm{x})}{\mathrm{H}}=\frac{\mathrm{M}(\mathrm{x})}{\mathrm{E}\mathrm{I}}$$

(12)

Thus,

$$\mathrm{M}\left(\mathrm{x}\right)=\mathrm{E}\mathrm{I}\frac{{\mathsf{\epsilon }}_{\mathrm{m}\mathrm{y}1}(\mathrm{x})-{\mathsf{\epsilon }}_{\mathrm{m}\mathrm{y}2}(\mathrm{x})}{\mathrm{H}}$$

(13)

If the load distribution ${\mathrm{q}}_{\mathrm{i},\mathrm{y}}^{\prime }$ of the conjugate beam equals $\frac{{\mathsf{\epsilon }}_{\mathrm{m}\mathrm{y}1}-{\mathsf{\epsilon }}_{\mathrm{m}\mathrm{y}2}}{\mathrm{H}}$, then the bending moment distribution of the conjugate beam is exactly the same as the vertical deformation distribution ${\mathsf{\nu }}_{\mathrm{y}}$(x) of the actual beam. Assuming that the length of each sensing unit is equal, it can be denoted as L. Based on the balance principle, the support reaction force at the left fulcrum of the conjugate beam can be determined as follows:

$$\overline{\mathrm{R}}=\frac{L}{n}{\sum }_{i=1}^n{q}_{i,y}^{\prime }(n-i+0.5).$$

(14)

Therefore, the bending moment of the middle point of each element of the conjugate beam, that is, the vertical displacement of the same point in the actual beam, can be expressed as:

$${\nu }_{i,y}=L^2{\sum }_{j=1}^i{q}_{j,y}^{\prime }(i-j)+\frac{q_{i,y}^{\prime }{\mathrm{L}}^2}{8}-\overline{\mathrm{R}}L(\mathrm{i}-0.5)(i=1,2,\cdots ,n)$$

(15)

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/m³, 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/m³.

The load acting on the structure was calculated with a “load-structure model”. The specific load calculation involved (1) roof surface water pressure ${\mathit{p}}_{\mathbf{1}}$ = 293.03 kPa, (2) floor surface water pressure ${\mathit{p}}_{\mathbf{2}}$ = 375.078 kPa, (3) covering pressure $\mathit{q}$ = −22.5 kPa, (4) earth pressure on the top of the side wall ${\mathit{e}}_{\mathit{t}}$ = −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 ${\mathit{e}}_{\mathit{b}}$ = 14.189 kPa.

As shown in Figure 3b, the comprehensive calculation results for the pressure on the section are as follows: $W_0=p_1+q=315.53 \mathrm{k}\mathrm{P}\mathrm{a}$, $W_1=p_2=375.078 \mathrm{k}\mathrm{P}\mathrm{a}$, $E_1=p_1+e_t=293.03 \mathrm{k}\mathrm{P}\mathrm{a}$, $E_2=p_2+e_b=389.267 \mathrm{k}\mathrm{P}\mathrm{a}$. 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 model were fixed. The soil mass was simulated using a total of 1452 solid elements and 9900 solid elements were used to simulate the tunnel structure. In the axial direction, the tunnel structure was discretized into 60 equal units connected by 61 nodes, each with a length of 1 m (Figure 4). The material properties of the soil layers and tunnel structure are listed in

Table 1. Material properties of the soil layers and tunnel structure.

	Density (kg/m3)	E (kPa)	Poisson’s Ratio	Cohesive Force (kPa)	Internal Friction Angle (°)	Expansion Angle (°)
Soil	2370	1.909 × 106	0.30	300	32.5	2.5
Structure	2500	3.45 × 104	0.20	_	_	_

.

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 με, and the measurement range is between −600 με and 3000 με. 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.

4.1. Coupled Strain Separation

The coupled strain time history data are collected by the long-gauge FBG sensors for four load cases. The coupled strains were then decoupled to obtain vertical bending strains for the tunnel unit via the proposed method. The results are shown in Figure 6.

Figure 6 illustrates the separated vertical bending strain along the tunnel in case C3. Focusing on point 1 on the section as the research object, Figure 6a,b shows the distribution of measured strain and vertical bending strain, respectively. The vertical bending strain is noticeably smaller in comparison to the measured strain. Moreover, the vertical strain distribution estimated from the measured strain is in close accordance with the actual values.

To further evaluate the accuracy of the estimated vertical bending strain, the separated strain was compared to the actual strain and the relative error (RE) was calculated. The results are shown in Figure 7.

Figure 7a–d shows the REs of separated strains in all four cases. Analyzing the RE values for the separated strain responses involved determining the maximum absolute value of RE (MAVRE):

$$\mathrm{M}\mathrm{A}\mathrm{V}\mathrm{R}\mathrm{E}=\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|\mathrm{R}\mathrm{E}\right|\right)$$

(16)

In all four cases, the MAVRE of the estimated vertical bending strain predominantly occurred on the unit near the support. Most did not exceed 0.5%. As shown in Figure 7a–d, the difference between the maximum and minimum values was generally maintained within a range of 0.3%. This implies that the proposed method ensures a high level of accuracy in calculating the vertical bending strain for all tunnel units, irrespective of the load conditions.

4.2. Vertical Displacement Estimation

Next, the separated vertical bending strain was used to calculate the vertical deformation distributions of the tunnel (Figure 8).

Figure 8 demonstrates the identified vertical displacement in cases C1 and C4. Figure 8a,b provides a comprehensive display of the vertical displacement throughout the tunnel with a focus on point 1. The traditional CBM was also used to perform vertical displacement identification using the measured coupled strain under the same simulation case conditions to demonstrate the advantages of the proposed method. As shown in Figure 8a,b, the displacement identified by CBM differed markedly from the actual value, whereas the computed result from the proposed method aligned closely with the true displacement.

Table 2. Identified vertical displacement (mm) in case C3.

Unit	PM	FEM	Unit	PM	FEM
1	0.14	0.16	16	2.09	2.11
2	0.30	0.32	17	2.18	2.19
3	0.46	0.48	18	2.26	2.27
4	0.61	0.63	19	2.34	2.35
5	0.76	0.78	20	2.41	2.42
6	0.91	0.92	21	2.47	2.48
7	1.05	1.07	22	2.53	2.54
8	1.18	1.20	23	2.58	2.59
9	1.32	1.33	24	2.63	2.63
10	1.44	1.46	25	2.66	2.67
11	1.57	1.58	26	2.69	2.70
12	1.68	1.70	27	2.72	2.72
13	1.79	1.81	28	2.74	2.74
14	1.90	1.91	29	2.75	2.75
15	2.00	2.01	30	2.75	2.75

shows the vertical displacement of 30 units identified by the proposed method and from the FEM for case C3, where the resolution of identified vertical displacement reaches 0.1 mm with certain units at the mid-span achieving a resolution of 0.01 mm. The REs of the identified vertical displacement in the four cases were also calculated, as shown in Figure 9.

Figure 9a–d, respectively, show the REs of the identified vertical displacement in all four cases. The maximum REs for the identified vertical displacement were all below 4%, while the minimum REs were all above −5%. Both the maximum and minimum REs occurred in the units at either end of the tunnel, where the strain is relatively small. Except for a few units near the ends of the tunnel, REs throughout the units were mostly within ±2%. These results confirm the reliability of the proposed method for vertical displacement measurement as the identification outcomes remain consistent regardless of the load distribution.

4.3. Noise Effects

The effects of different noise levels on the proposed method were also evaluated. Normally distributed random noise with zero mean and unit standard deviation was added to simulate the effect of measurement noise:

$$\ddot{\epsilon }={\epsilon }_c+E_{\pi }{N}_{oise}std\left({\epsilon }_c\right)$$

(17)

where $\ddot{\epsilon }$ and ${\epsilon }_c$ are the simulated noisy and original responses, respectively; $E_{\pi }$ is the noise level; $N_{oise}$ is a standard normal distribution vector with zero mean and unit standard deviation; $std\left({\epsilon }_c\right)$ denotes the standard deviation of the original response.

Noise effects of 2%, 4%, and 6% were added to the simulated monitoring data for case C4 (Section 4.1). The vertical bending strain was first separated from the coupled strain data using the proposed method; then, the RE was calculated by comparing the separated vertical bending strain with its true value. The results are shown in Figure 10.

Figure 10 shows the RE for the separated vertical bending strain. At noise levels of 2%, 4%, and 6%, the MAVREs were obtained as listed in

Table 3. MAVPEs (%) of estimated strain at different noise levels.

Noise	2%	4%	6%
MAVPE	1.37	3.23	5.17

. The units wherein MAVREs occurred were 3, 2, and 2, with a notable concentration near the support, where the responses were relatively small. However, for units closer to the middle span, the RE increased slightly with the noise level, with most remaining below 2%. At the 6% noise level, the absolute Res of all units (except for one exceeding 5%) remained at a low level in the estimated vertical bending strain (Figure 10c). This demonstrates that the accuracy of the vertical bending strain estimated by the proposed method only marginally decreases as the noise level increases, while the error remains within an acceptable range.

Figure 11a–c shows the REs of the identified vertical displacement at the three noise levels. Fluctuations in the RE curve gradually intensified as the noise level increased. The MAVRE for the identified displacement across the entire tunnel was highest for the units with small responses at both ends, while the accuracy of identified displacement in the middle span of the tunnel was consistently high.

Table 4. MAVPEs (%) of identified vertical displacement at different noise levels.

Noise	2%	4%	6%
MAVPE	3.79	3.31	3.30

lists the MAVREs for the identified vertical displacement at these three noise levels, none of which exceeded 5%. Thus, the proposed method exhibits robustness to various noise levels, effectively enabling identification of the vertical deformation distribution of the tunnel under various noise conditions.

4.4. Sensor Layout

In the process of field-monitoring tunnels, budgetary constraints and practical conditions often limit the deployment of sensors within the tunnel itself. As discussed in this section, we explored four distinct sensor layout schemes (denoted as Schemes 1–4) to verify the effectiveness and robustness of the proposed method for estimating the vertical displacement in a tunnel under realistic field monitoring conditions.

Schemes 1 and 2 entail sensor arrangements involving five monitoring units, while Schemes 3 and 4 focus on sensor arrangements with seven monitoring units. For Schemes 1 and 3, three sensors were arranged on each monitoring unit and the local sensor arrangement is as shown in Section II of Figure 12. For Schemes 2 and 4, three sensors were arranged on part of the monitoring units and only the top sensor (Point 1) was reserved for the remaining monitoring units, as also illustrated in Figure 12. Other details regarding the four sensor layouts are provided in

Table 5. Sensor layout schemes.

Scheme	Number of Sensors	Monitoring Unit	Local Sensor Layout
			Monitoring Section I	Monitoring Section II
1	15	1 15 30 45 60	1 15 30 45 60
2	11	1 15 30 45 60	1 30 60	15 45
3	21	1 10 20 30 40 50 60	1 10 20 30 40 50 60
4	13	1 10 20 30 40 50 60	1 20 40 60	10 30 50

.

The load from case C3 (Section 4.1) was applied in this analysis. Initially, the coupled strain on the corresponding monitoring unit was extracted according to different sensor layout schemes. For schemes 1 and 3, the responses of the unmonitored units were then calculated via curve interpolation using the monitoring unit, then strain separation and displacement estimation were carried out according to the proposed method. As there was only one monitoring point on certain units in schemes 2 and 4, the strain data for points 2 and 3 were firstly obtained on each monitoring unit with reference to the sensor layout scheme. Next, the strain responses of all units on the investigated tunnel were obtained via interpolation method using the measured strain. Finally, strain separation and displacement estimation were conducted using the proposed method.

Figure 13 shows the REs between the separated vertical bending strain and the actual strain with different sensor layout schemes.

Table 6. MAVPEs (%) of estimated strain for different sensor layout schemes.

Scheme	1	2	3	4
MAVPE	2.60	7.58	0.22	6.32

lists the MAVPEs of estimated strains for the different schemes. By comparing scheme 1 to scheme 3, the sensor arrangement containing seven monitoring sections appears to ensure that the RE of strain separation remains stable throughout the tunnel and does not exceed 1%; the sensor arrangement containing five monitoring sections causes the RE of strain separation to fluctuate slightly throughout the tunnel, though the MAVRE still does not exceed 3%. The MAVRE is slightly smaller in the case of scheme 4 compared to scheme 2, indicating that the separated strain of scheme 4 is closer to the true value than that of scheme 2. Although the absolute RE values of a few units in scheme 4 exceed 5%, these units are located near the support and thus have a small response, so they would not substantially impact the displacement identification in practice.

Figure 14 shows the REs between the estimated displacement and actual displacement for the four sensor layout schemes. The MAVPEs for the identified vertical displacement under various schemes are listed in

Table 7. MAVPEs (%) of identified vertical displacement for different sensor layout schemes.

Scheme	1	2	3	4
MAVPE	5.12	7.73	4.54	6.52

. The MAVPEs for schemes 1 and 3 are consistently maintained at approximately 5%, while the MAVPEs for schemes 2 and 4 fall below 8%. However, the units experiencing MAVPE are primarily situated near the supports, where the strain responses are relatively small. In contrast, the REs for units located in the mid-span region are generally contained within 3%; their strain responses exert a significant impact on the tunnel structure. The differences in the maximum and minimum REs between schemes 1 and 3 are all less than 1%, which indicates that the identified displacements under the two schemes are basically consistent.

Moreover, the MAVRE in scheme 4 is smaller than the corresponding value in scheme 2. Although a small number of units in schemes 4 and 2 exceed 5%, with only two units in scheme 4 to do so, these units are consistently located near the left support and exhibit small responses. Despite scheme 4 incorporating two additional sensors compared to scheme 2, the identified displacement is marginally superior to that of scheme 2.

5. Conclusions

This paper proposed a novel method for measuring vertical displacement in tunnels based on long-gauge strain sensing. Its effectiveness was validated through a series of numerical simulations. The main conclusions of this study can be summarized as follows.

(1)

The proposed method effectively addresses the challenge of accurately measuring vertical displacement in the presence of structural coupling deformation. This is achieved by separating the bending strain from the measured strain to estimate the vertical deformation distribution of the tunnel structure.

(2)

The innovative sensor layout adopted in the monitoring section enables the vertical displacement measurement of the tunnel. This sensor layout not only realizes the decoupling of the measured strain, but also guarantees accurate displacement estimations for the tunnel.

(3)

Numerical examples successfully confirmed the accurate identification of vertical displacement in a tunnel under different loading conditions, with a maximum error of less than 5%, satisfying the requirements of practical applications. This method also achieves a resolution of 0.1 mm for the identified vertical displacement, with certain units at the mid-span achieving a resolution of 0.01 mm. Fluctuating noise levels and various sensor layout schemes were tested to find that they do not degrade the accuracy of the proposed method.

A series of calculation results validated the suitability of the proposed method, thus offering a valuable reference for the analysis of data collected through field monitoring. In the future, our primary focus will be analyzing a real-world damaged tunnel structure. This is crucial, as the presence of damage in the tunnel structure can potentially impact the proposed displacement monitoring method.