# Simultaneous Estimation of Submerged Floating Tunnel Displacement and Mooring Cable Tension through FIR Filter-Based Strain and Acceleration Fusion

Authors: Zhanxiong Ma; Jaemook Choi; Jinho Jang; Ohjun Kwon; Hoon Sohn
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

A submerged floating tunnel (SFT) is a tunnel structure that floats approximately 50 m below the water surface and is anchored to the seabed by mooring cables. It is a new alternative to conventional bridges and immersed tunnels. There have been ongoing efforts to construct SFTs worldwide, and the integrity of these SFTs needs to be monitored throughout their lifespan. This study simultaneously estimated the tunnel displacement and mooring cable tension force using acceleration and strain measurements to assess the integrity of an SFT. First, strain measurements were transformed to displacement using simplified mode shapes and mode-scaling factors, which did not require the true mode shapes of the SFT. The mode-scaling factors were automatically estimated using initial strain and acceleration measurements. Then, the strain-based displacement was combined with the acceleration measurement using a finite impulse response filter to improve the displacement estimation accuracy. In addition, the tension force of a mooring cable was estimated from the displacement at the connection between the tunnel and mooring cable. The feasibility of the proposed technique was examined through a series of numerical simulations and laboratory tests on an 8 m-long aluminum SFT mock-up structure.


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## Body

## 1. Introduction

A submerged floating tunnel (SFT) is a tunnel structure floating approximately 50 m below the water surface and anchored to the seabed by mooring cables, as shown in Figure1. The SFT is an attractive alternative to conventional bridges or immersed tunnels because its construction is not limited by the span length or depth of the ocean floor [1]. So far, no real-world SFT has been constructed, but there are ongoing efforts to build SFTs [2–5]. The development of monitoring systems to continuously evaluate the integrity of these SFTs is necessary. In particular, the measurements of the tunnel displacement and mooring cable tension force are critical for the structural integrity assessment of an SFT.Figure 1
A conceptual sketch of a submerged floating tunnel (SFT) anchored to the seabed by mooring cables and sensor configuration of the proposed technique.Several techniques are available for direct displacement measurements, utilizing linear variable differential transformers (LVDT) [6], real-time kinematic global navigation satellite systems (RTK-GNSS) [7], vision-based systems [8], and radar systems [9, 10]. Indirect displacement estimation techniques using accelerometers, inclinometers, and strain sensors have been developed recently as well [11–13]. Another trend for structural displacement estimation involves combining multiple types of measurements to improve the displacement measurement accuracy and reliability. Examples include combining RTK-GNSS and accelerometer [14], strain gauge and accelerometer [15, 16], vision camera and accelerometer [17, 18], and millimeter-wave radar and accelerometer [19]. Obviously, GNSS, radar, and vision-based techniques cannot operate in underwater environments, and the fusion of strain gauges and accelerometers seems to be promising for SFT displacement monitoring.In addition to displacement, monitoring of the mooring cable tension force is also essential because mooring cables are major load-carrying members. Several cable tension force estimation techniques have been proposed, utilizing load cells [20], strain gauges [21], accelerometers [22, 23], electromagnetic (EM) sensors [24, 25], and eddy current sensors [26]. However, all these discrete sensors must be physically placed on each cable, and such an installation can be time-consuming and labor-intensive. Although several attempts have been made to develop computer-vision techniques for noncontact tension force estimation [27, 28], vision techniques cannot work in underwater environments.This study simultaneously estimates the tunnel displacement and mooring cable tension force using acceleration and strain measurements. The sensor configuration for the proposed technique is shown in Figure1. Strain gauges are installed at the mid-span locations of each tunnel segment (x=xm,i,i=1,⋯,N), and bidirectional accelerometers are installed at locations where the displacement needs to be estimated. Figure 2 shows the flowchart of the proposed technique. First, the unknown mode-scaling factors necessary for the strain-displacement transformation (derived in Section 2) are estimated using the initial strain and acceleration measurements (less than 1 min) (Section 3.1). Second, the strain measurements after the first step are transformed to strain-based displacement using the derived transformation equation and estimated mode-scaling factors. Third, a finite impulse response (FIR) filter combines the strain-based displacement with acceleration measurements to improve displacement estimation accuracy (Section 3.2). Finally, the tension force of the mooring cable is estimated using the displacement estimated at the connection point of the mooring cable with the tunnel (Section 3.2.2). The performance of the proposed technique is examined in Section 4 through a series of numerical simulations, and in Section 5, a laboratory test on an 8 m-long aluminum SFT mock-up structure is presented. Concluding remarks are provided in Section 6. Note that no real-world SFT construction has been realized; an SFT structure form, commonly used in previous studies [1–3] (as shown in Figure 1), is adopted in this study.Figure 2
Flowchart of the proposed technique for simultaneous estimation of tunnel displacement and mooring cable tension using acceleration and strain measurements.This study has the following contributions: (1) the displacement of an SFT is estimated by combining strain and acceleration measurements without knowing the true mode shapes of the SFT; (2) the tension force of a mooring cable is also estimated from the displacement at the connection point between the tunnel and the cable; and (3) the proposed displacement and tension force estimation technique are experimentally validated on an 8 m-long aluminum SFT structure.


## 2. Derivation of Strain-Displacement Transformation Relationship for SFTs with Simplified Mode Shapes

### 2.1. Simplified Model of SFTs

Previous studies have shown that an SFT can be simplified as a simple beam on an elastic foundation if the stiffness ratio is less than 0.05 [29]:(1)KDx324EI<0.05,where K denotes the axial stiffness of the mooring cables, Dx is the distance between two adjacent mooring cables in the x direction, and EI denotes the bending stiffness of the tunnel. When equation (1) is satisfied, the displacement mode shapes of a simply supported beam (SSB) (ψ¯jx), hereafter referred to as the simplified displacement mode shapes, can approximate the true displacement mode shapes of an SFT (ψjx). Here, subscript j denotes the jth mode. These simplified displacement mode shapes have been commonly used to analyze the dynamic responses of an SFT under seismic, vehicle, and impact loadings [30–32].In practice, it is difficult to measure the structural displacement, especially in underwater environments. To tackle this issue, the strain is measured and transformed to displacement using a strain-displacement transformation relationship, which requires the strain mode shapes. Figure3 shows the differences between the strain mode shapes of the SSB φ¯jx and the SFT φjx. However, if only the strain mode shapes at the mid-span locations are of interest (φjx at xm,1, xm,2, …, xm,N), φjx can be easily approximated by scaling φ¯jx with a mode-scaling factor αj. Note that as the mode order increases, the differences between φ¯jx and φjx decrease, as well as the mode-scaling factor value (Figure 3). Here, the strain mode shapes of the SFT are from one of the models investigated in numerical validation section.Figure 3
Comparison of the true and simplified strain mode shapes of an SFT.


### 2.2. Strain-Displacement Transformation

Figure1 depicts an example of an SFT with strain gauges installed at N mid-span locations along the tunnel x=xm,i,i=1,⋯,N. At the cross-section of each location, three strain gauges (S1–S3) are attached at three different points along the vertical (z) and horizontal (y) directions. The measurements from strain gauges S1, S2, and S3 are denoted as εk1, εk2, and εk3. Here, subscript k denotes the kth time step. Based on a mode superposition algorithm [16], the strain measurements εk1 and εk2 can be transformed into vertical displacement ukzx as follows:(2)ukzx=1hψ1zx⋮ψNzxTφ1zxm,1…φNzxm,1⋮⋱⋮φ1zxm,N…φNzxm,N−1∆εk1,2xm,1⋮∆εk1,2xm,N,(3)∆εk1,2xm,i=εk1xm,i−εk2xm,i,i=1,⋯,N,where h denotes the vertical distance between strain gauges S1 and S2. Using the simplified strain and displacement mode shapes, equation (2) becomes(4)ukzx=1hψ¯1x⋮ψ¯NxT1α1z⋱1αNzφ¯ixm,1…φ¯Nxm,1⋮⋱⋮φ¯ixm,N…φ¯Nxm,N−1∆εk1,2xm,1⋮∆εk1,2xm,N.Equation (4) involves N mode-scaling factors, and their estimation is discussed in Section 3.1. Similarly, the horizontal displacement (uky) can be estimated from the strain measurements εk1 and εk3.


## 2.1. Simplified Model of SFTs

Previous studies have shown that an SFT can be simplified as a simple beam on an elastic foundation if the stiffness ratio is less than 0.05 [29]:(1)KDx324EI<0.05,where K denotes the axial stiffness of the mooring cables, Dx is the distance between two adjacent mooring cables in the x direction, and EI denotes the bending stiffness of the tunnel. When equation (1) is satisfied, the displacement mode shapes of a simply supported beam (SSB) (ψ¯jx), hereafter referred to as the simplified displacement mode shapes, can approximate the true displacement mode shapes of an SFT (ψjx). Here, subscript j denotes the jth mode. These simplified displacement mode shapes have been commonly used to analyze the dynamic responses of an SFT under seismic, vehicle, and impact loadings [30–32].In practice, it is difficult to measure the structural displacement, especially in underwater environments. To tackle this issue, the strain is measured and transformed to displacement using a strain-displacement transformation relationship, which requires the strain mode shapes. Figure3 shows the differences between the strain mode shapes of the SSB φ¯jx and the SFT φjx. However, if only the strain mode shapes at the mid-span locations are of interest (φjx at xm,1, xm,2, …, xm,N), φjx can be easily approximated by scaling φ¯jx with a mode-scaling factor αj. Note that as the mode order increases, the differences between φ¯jx and φjx decrease, as well as the mode-scaling factor value (Figure 3). Here, the strain mode shapes of the SFT are from one of the models investigated in numerical validation section.Figure 3
Comparison of the true and simplified strain mode shapes of an SFT.


## 2.2. Strain-Displacement Transformation

Figure1 depicts an example of an SFT with strain gauges installed at N mid-span locations along the tunnel x=xm,i,i=1,⋯,N. At the cross-section of each location, three strain gauges (S1–S3) are attached at three different points along the vertical (z) and horizontal (y) directions. The measurements from strain gauges S1, S2, and S3 are denoted as εk1, εk2, and εk3. Here, subscript k denotes the kth time step. Based on a mode superposition algorithm [16], the strain measurements εk1 and εk2 can be transformed into vertical displacement ukzx as follows:(2)ukzx=1hψ1zx⋮ψNzxTφ1zxm,1…φNzxm,1⋮⋱⋮φ1zxm,N…φNzxm,N−1∆εk1,2xm,1⋮∆εk1,2xm,N,(3)∆εk1,2xm,i=εk1xm,i−εk2xm,i,i=1,⋯,N,where h denotes the vertical distance between strain gauges S1 and S2. Using the simplified strain and displacement mode shapes, equation (2) becomes(4)ukzx=1hψ¯1x⋮ψ¯NxT1α1z⋱1αNzφ¯ixm,1…φ¯Nxm,1⋮⋱⋮φ¯ixm,N…φ¯Nxm,N−1∆εk1,2xm,1⋮∆εk1,2xm,N.Equation (4) involves N mode-scaling factors, and their estimation is discussed in Section 3.1. Similarly, the horizontal displacement (uky) can be estimated from the strain measurements εk1 and εk3.


## 3. Development of Displacement and Tension Force Estimation Technique for SFTs

This section proposes a simultaneous displacement and cable tension estimation technique for SFTs, using strain and acceleration measurements. As shown in Figure2, the proposed technique consists of two steps: (1) automated mode-scaling factor estimation (Section 3.1) and (2) continuous displacement and mooring cable tension force estimation (Section 3.2.1). Note that the mode-scaling factor and displacement estimation steps are explained only for vertical displacement, but these steps can be easily extended to estimate the horizontal displacement as well.


### 3.1. Automated Mode-Scaling Factor Estimation

An automated algorithm is proposed in this section to estimate the values of the unknown mode-scaling factors in equation (4), which are necessary for the strain-displacement transformation. Note that out of N unknown mode-scaling factors, only the first P (<N) scaling factors are estimated, and the remaining high-order scaling factor values are set to 1. This approximation can be justified on the basis of (1) the difference between the simplified and true mode shapes and (2) the contribution of the higher modes to the structural displacement, where both decrease as the mode order increases (Figure 3). It should be noted that because at least P equations are needed for the estimation of P mode-scaling factors, equation (4) is extended to include displacement estimation at P locations (xd,i,1≤i≤P):(5)ukzxd,1⋮ukzxd,P=ψ¯Fullh1α1z⋱1αPz1⋱⋱1φ¯1xm,1…φ¯Nxm,1⋮⋱⋮φ¯1xm,N…φ¯Nxm,N−1∆εk1,2xm,1⋮∆εk1,2xm,N,(6)ψ¯Full=ψ¯Lowψ¯High=ψ¯1xd,1⋯ψ¯Pxd,1⋮⋱⋮ψ¯1xd,P⋯ψ¯Pxd,Pψ¯P+1xd,1⋯ψ¯Nxd,1⋮⋱⋮ψ¯P+1xd,P⋯ψ¯Nxd,P.


#### 3.1.1. Transformation Equation Decoupling

First, equation (5) is decoupled into P independent equations before estimating the values of the mode-scaling factors. The strain-based modal responses q¯j,ks, 1≤j≤P are defined as follows:(7)q¯1,ks⋮q¯N,ks=1hφ¯1xm,1…φ¯Nxm,1⋮⋱⋮φ¯1xm,N…φ¯Nxm,N−1∆εk1,2xm,1⋮∆εk1,2xm,N.The following equation is obtained by introducing equation (7) into equation (5):(8)ukz=ukzxd,1⋮ukzxd,P,=ψ¯Low1α1z⋱1αPzq¯1,ks⋮q¯P,ks+ψ¯Highq¯P+1,ks⋮q¯N,ks.Equation (8) can be further rewritten as follows:(9)q1,ku⋮qP,ku=1α1z⋱1αPzq¯1,ks⋮q¯P,ks,=ψ¯Low−1ukz−ψ¯Highq¯P+1,ks⋮q¯N,ks,where qj,ku denotes the jth modal response decomposed from ukz after removing the high-order >P modal responses. Finally, P independent equations are obtained, and each of them includes only one mode-scaling factor.(10)qj,ku=q¯j,ksαjz,j=1,⋯,P.


#### 3.1.2. Mode-Scaling Factor Estimation

Assuming that accelerometers are installed atP locations and strain gauges are installed at N locations, a flowchart of the proposed mode-scaling factor estimation algorithm is presented in Figure 4. Note that the strain gauges must be installed at the mid-span locations of each tunnel segment, whereas the accelerometers can be installed at arbitrary locations along the tunnel.Figure 4
Flowchart of proposed mode-scaling factor estimation algorithm.First,∆εk1,2xm,i, for i=1,⋯,N, are computed from strain measurements using equation (3) and high-pass filtered. Subsequently, the high-pass filtered ∆εk1,2xm,i are decomposed into strain-based modal responses q¯i,ks using equation (7). The strain-based modal responses are divided into two parts: (1) high-order j>P and (2) low-order j≤P.Second, high-frequency displacements are estimated atP locations by double integrating and high-pass filtering acceleration measurements. Then, acceleration-based low-order modal responses qj,ku,1≤j≤P are estimated from the acceleration-based high-frequency displacements and high-orderq¯j,ks,j>Pstrain-based modal responses using equation (9).Assuming that both the acceleration and strain are measured for a short time (less than 1 min),qj,ku and q¯j,ks are obtained at multiple time steps. Therefore, αjz can be estimated as the slope of the q¯j,ks vs. qj,ku plot using least-squares regression. The purpose of high-pass filtering is to remove large low-frequency drifts in the acceleration-based displacements. The cutoff frequency of the filter should, therefore, be sufficiently high to remove drifts and should be lower than the first resonance frequency of the SFT. Note that an identical high-pass filter should be applied to both strain and acceleration measurements.


### 3.2. Continuous Displacement and Mooring Cable Tension Force Estimation

After estimating the mode-scaling factors using the initial acceleration and strain measurements, the displacement and mooring cable tension force can be continuously monitored.


#### 3.2.1. FIR Filter-Based Displacement Estimation

Displacement estimated from acceleration measurement using the double integration has a low-frequency drift caused by unknown initial conditions and measurement noises. A high-pass filter can be used to remove the low-frequency drift [11], but it also removes the low-frequency structural displacement. Displacement estimated from strain measurement usually has low accuracy owing to relatively large noise levels in strain measurement [12, 13]. An FIR filter was adopted in this study to estimate the SFT displacement by fusing strain and acceleration measurements, and the final displacement is estimated as a combination of low-frequency displacement from strain measurements and high-frequency displacement from acceleration measurements, which provides better accuracy than strain-based displacement. Figure 5 presents a flowchart of the proposed FIR filter-based technique, assuming that strain gauges are installed at N mid-span locations and an accelerometer is installed at the target location where displacement needs to be estimated.Figure 5
Flowchart of FIR filter-based continuous displacement estimation by combining strain and acceleration measurements.First, multiple strain measurements are transformed into displacement at the target location using equation (4) and the estimated mode-scaling factors. The strain-based displacement is then filtered using a low-pass filter CL to obtain low-frequency displacement, while the high-frequency displacement is obtained from double integration and the high-pass filtered acceleration measurements, where double integration and high-pass filtering are simultaneously performed by CH. The cutoff frequencies of the low-pass and high-pass filters are set to the first resonance frequency of the SFT. More details on CL and CH can be found in the Appendix and the study by Ma et al. [16]. The final displacement is obtained by combining the high-frequency displacement obtained from the acceleration measurement and the low-frequency displacement obtained from the strain measurements.


#### 3.2.2. Mooring Cable Tension Force Estimation

The tension force of the mooring cable is estimated from both the horizontal and vertical displacements at the connection point between the tunnel and mooring cable. For the SFT shown in Figure6, the following equation can be derived from the geometric relationship between the initial t=0 and current t=k∆t states of the SFT:(11)y0+uky−R2+z0+ukz2=Lkl;y0+uky+R−Dy/22+z0+ukz2=Lkr,where R denotes the outer radius of the tunnel and Dy denotes the distance between two anchoring points along the y-direction. y0,z0 denotes the initial position of the center of the tunnel cross-section. Lkl and Lkr denote the lengths of the left and right cables at the kth time step, respectively. Lkl and Lkr are related to the initial lengths L0l and L0r, axial stiffness (K), and tension forces TkL and Tkr of the two cables, as follows:(12)Lkl=1+TklKL0l;Lkr=1+TkrKL0r.Figure 6
Relationship between the tunnel displacement and the anchoring positions of two mooring cables.By combining equations (11) and (12), the tension forces can be estimated as follows:(13)TkL=y0+uky−R2+z0+ukz2−L0lKL0l,Tkr=y0+uky+R−Dy/22+z0+ukz2−L0rKL0r.Note that the mooring cables are assumed to be straight without any sag and are always in tension because of the large initial tension.


## 3.1. Automated Mode-Scaling Factor Estimation

An automated algorithm is proposed in this section to estimate the values of the unknown mode-scaling factors in equation (4), which are necessary for the strain-displacement transformation. Note that out of N unknown mode-scaling factors, only the first P (<N) scaling factors are estimated, and the remaining high-order scaling factor values are set to 1. This approximation can be justified on the basis of (1) the difference between the simplified and true mode shapes and (2) the contribution of the higher modes to the structural displacement, where both decrease as the mode order increases (Figure 3). It should be noted that because at least P equations are needed for the estimation of P mode-scaling factors, equation (4) is extended to include displacement estimation at P locations (xd,i,1≤i≤P):(5)ukzxd,1⋮ukzxd,P=ψ¯Fullh1α1z⋱1αPz1⋱⋱1φ¯1xm,1…φ¯Nxm,1⋮⋱⋮φ¯1xm,N…φ¯Nxm,N−1∆εk1,2xm,1⋮∆εk1,2xm,N,(6)ψ¯Full=ψ¯Lowψ¯High=ψ¯1xd,1⋯ψ¯Pxd,1⋮⋱⋮ψ¯1xd,P⋯ψ¯Pxd,Pψ¯P+1xd,1⋯ψ¯Nxd,1⋮⋱⋮ψ¯P+1xd,P⋯ψ¯Nxd,P.


### 3.1.1. Transformation Equation Decoupling

First, equation (5) is decoupled into P independent equations before estimating the values of the mode-scaling factors. The strain-based modal responses q¯j,ks, 1≤j≤P are defined as follows:(7)q¯1,ks⋮q¯N,ks=1hφ¯1xm,1…φ¯Nxm,1⋮⋱⋮φ¯1xm,N…φ¯Nxm,N−1∆εk1,2xm,1⋮∆εk1,2xm,N.The following equation is obtained by introducing equation (7) into equation (5):(8)ukz=ukzxd,1⋮ukzxd,P,=ψ¯Low1α1z⋱1αPzq¯1,ks⋮q¯P,ks+ψ¯Highq¯P+1,ks⋮q¯N,ks.Equation (8) can be further rewritten as follows:(9)q1,ku⋮qP,ku=1α1z⋱1αPzq¯1,ks⋮q¯P,ks,=ψ¯Low−1ukz−ψ¯Highq¯P+1,ks⋮q¯N,ks,where qj,ku denotes the jth modal response decomposed from ukz after removing the high-order >P modal responses. Finally, P independent equations are obtained, and each of them includes only one mode-scaling factor.(10)qj,ku=q¯j,ksαjz,j=1,⋯,P.


### 3.1.2. Mode-Scaling Factor Estimation

Assuming that accelerometers are installed atP locations and strain gauges are installed at N locations, a flowchart of the proposed mode-scaling factor estimation algorithm is presented in Figure 4. Note that the strain gauges must be installed at the mid-span locations of each tunnel segment, whereas the accelerometers can be installed at arbitrary locations along the tunnel.Figure 4
Flowchart of proposed mode-scaling factor estimation algorithm.First,∆εk1,2xm,i, for i=1,⋯,N, are computed from strain measurements using equation (3) and high-pass filtered. Subsequently, the high-pass filtered ∆εk1,2xm,i are decomposed into strain-based modal responses q¯i,ks using equation (7). The strain-based modal responses are divided into two parts: (1) high-order j>P and (2) low-order j≤P.Second, high-frequency displacements are estimated atP locations by double integrating and high-pass filtering acceleration measurements. Then, acceleration-based low-order modal responses qj,ku,1≤j≤P are estimated from the acceleration-based high-frequency displacements and high-orderq¯j,ks,j>Pstrain-based modal responses using equation (9).Assuming that both the acceleration and strain are measured for a short time (less than 1 min),qj,ku and q¯j,ks are obtained at multiple time steps. Therefore, αjz can be estimated as the slope of the q¯j,ks vs. qj,ku plot using least-squares regression. The purpose of high-pass filtering is to remove large low-frequency drifts in the acceleration-based displacements. The cutoff frequency of the filter should, therefore, be sufficiently high to remove drifts and should be lower than the first resonance frequency of the SFT. Note that an identical high-pass filter should be applied to both strain and acceleration measurements.


## 3.1.1. Transformation Equation Decoupling

First, equation (5) is decoupled into P independent equations before estimating the values of the mode-scaling factors. The strain-based modal responses q¯j,ks, 1≤j≤P are defined as follows:(7)q¯1,ks⋮q¯N,ks=1hφ¯1xm,1…φ¯Nxm,1⋮⋱⋮φ¯1xm,N…φ¯Nxm,N−1∆εk1,2xm,1⋮∆εk1,2xm,N.The following equation is obtained by introducing equation (7) into equation (5):(8)ukz=ukzxd,1⋮ukzxd,P,=ψ¯Low1α1z⋱1αPzq¯1,ks⋮q¯P,ks+ψ¯Highq¯P+1,ks⋮q¯N,ks.Equation (8) can be further rewritten as follows:(9)q1,ku⋮qP,ku=1α1z⋱1αPzq¯1,ks⋮q¯P,ks,=ψ¯Low−1ukz−ψ¯Highq¯P+1,ks⋮q¯N,ks,where qj,ku denotes the jth modal response decomposed from ukz after removing the high-order >P modal responses. Finally, P independent equations are obtained, and each of them includes only one mode-scaling factor.(10)qj,ku=q¯j,ksαjz,j=1,⋯,P.


## 3.1.2. Mode-Scaling Factor Estimation

Assuming that accelerometers are installed atP locations and strain gauges are installed at N locations, a flowchart of the proposed mode-scaling factor estimation algorithm is presented in Figure 4. Note that the strain gauges must be installed at the mid-span locations of each tunnel segment, whereas the accelerometers can be installed at arbitrary locations along the tunnel.Figure 4
Flowchart of proposed mode-scaling factor estimation algorithm.First,∆εk1,2xm,i, for i=1,⋯,N, are computed from strain measurements using equation (3) and high-pass filtered. Subsequently, the high-pass filtered ∆εk1,2xm,i are decomposed into strain-based modal responses q¯i,ks using equation (7). The strain-based modal responses are divided into two parts: (1) high-order j>P and (2) low-order j≤P.Second, high-frequency displacements are estimated atP locations by double integrating and high-pass filtering acceleration measurements. Then, acceleration-based low-order modal responses qj,ku,1≤j≤P are estimated from the acceleration-based high-frequency displacements and high-orderq¯j,ks,j>Pstrain-based modal responses using equation (9).Assuming that both the acceleration and strain are measured for a short time (less than 1 min),qj,ku and q¯j,ks are obtained at multiple time steps. Therefore, αjz can be estimated as the slope of the q¯j,ks vs. qj,ku plot using least-squares regression. The purpose of high-pass filtering is to remove large low-frequency drifts in the acceleration-based displacements. The cutoff frequency of the filter should, therefore, be sufficiently high to remove drifts and should be lower than the first resonance frequency of the SFT. Note that an identical high-pass filter should be applied to both strain and acceleration measurements.


## 3.2. Continuous Displacement and Mooring Cable Tension Force Estimation

After estimating the mode-scaling factors using the initial acceleration and strain measurements, the displacement and mooring cable tension force can be continuously monitored.


### 3.2.1. FIR Filter-Based Displacement Estimation

Displacement estimated from acceleration measurement using the double integration has a low-frequency drift caused by unknown initial conditions and measurement noises. A high-pass filter can be used to remove the low-frequency drift [11], but it also removes the low-frequency structural displacement. Displacement estimated from strain measurement usually has low accuracy owing to relatively large noise levels in strain measurement [12, 13]. An FIR filter was adopted in this study to estimate the SFT displacement by fusing strain and acceleration measurements, and the final displacement is estimated as a combination of low-frequency displacement from strain measurements and high-frequency displacement from acceleration measurements, which provides better accuracy than strain-based displacement. Figure 5 presents a flowchart of the proposed FIR filter-based technique, assuming that strain gauges are installed at N mid-span locations and an accelerometer is installed at the target location where displacement needs to be estimated.Figure 5
Flowchart of FIR filter-based continuous displacement estimation by combining strain and acceleration measurements.First, multiple strain measurements are transformed into displacement at the target location using equation (4) and the estimated mode-scaling factors. The strain-based displacement is then filtered using a low-pass filter CL to obtain low-frequency displacement, while the high-frequency displacement is obtained from double integration and the high-pass filtered acceleration measurements, where double integration and high-pass filtering are simultaneously performed by CH. The cutoff frequencies of the low-pass and high-pass filters are set to the first resonance frequency of the SFT. More details on CL and CH can be found in the Appendix and the study by Ma et al. [16]. The final displacement is obtained by combining the high-frequency displacement obtained from the acceleration measurement and the low-frequency displacement obtained from the strain measurements.


### 3.2.2. Mooring Cable Tension Force Estimation

The tension force of the mooring cable is estimated from both the horizontal and vertical displacements at the connection point between the tunnel and mooring cable. For the SFT shown in Figure6, the following equation can be derived from the geometric relationship between the initial t=0 and current t=k∆t states of the SFT:(11)y0+uky−R2+z0+ukz2=Lkl;y0+uky+R−Dy/22+z0+ukz2=Lkr,where R denotes the outer radius of the tunnel and Dy denotes the distance between two anchoring points along the y-direction. y0,z0 denotes the initial position of the center of the tunnel cross-section. Lkl and Lkr denote the lengths of the left and right cables at the kth time step, respectively. Lkl and Lkr are related to the initial lengths L0l and L0r, axial stiffness (K), and tension forces TkL and Tkr of the two cables, as follows:(12)Lkl=1+TklKL0l;Lkr=1+TkrKL0r.Figure 6
Relationship between the tunnel displacement and the anchoring positions of two mooring cables.By combining equations (11) and (12), the tension forces can be estimated as follows:(13)TkL=y0+uky−R2+z0+ukz2−L0lKL0l,Tkr=y0+uky+R−Dy/22+z0+ukz2−L0rKL0r.Note that the mooring cables are assumed to be straight without any sag and are always in tension because of the large initial tension.


## 3.2.1. FIR Filter-Based Displacement Estimation

Displacement estimated from acceleration measurement using the double integration has a low-frequency drift caused by unknown initial conditions and measurement noises. A high-pass filter can be used to remove the low-frequency drift [11], but it also removes the low-frequency structural displacement. Displacement estimated from strain measurement usually has low accuracy owing to relatively large noise levels in strain measurement [12, 13]. An FIR filter was adopted in this study to estimate the SFT displacement by fusing strain and acceleration measurements, and the final displacement is estimated as a combination of low-frequency displacement from strain measurements and high-frequency displacement from acceleration measurements, which provides better accuracy than strain-based displacement. Figure 5 presents a flowchart of the proposed FIR filter-based technique, assuming that strain gauges are installed at N mid-span locations and an accelerometer is installed at the target location where displacement needs to be estimated.Figure 5
Flowchart of FIR filter-based continuous displacement estimation by combining strain and acceleration measurements.First, multiple strain measurements are transformed into displacement at the target location using equation (4) and the estimated mode-scaling factors. The strain-based displacement is then filtered using a low-pass filter CL to obtain low-frequency displacement, while the high-frequency displacement is obtained from double integration and the high-pass filtered acceleration measurements, where double integration and high-pass filtering are simultaneously performed by CH. The cutoff frequencies of the low-pass and high-pass filters are set to the first resonance frequency of the SFT. More details on CL and CH can be found in the Appendix and the study by Ma et al. [16]. The final displacement is obtained by combining the high-frequency displacement obtained from the acceleration measurement and the low-frequency displacement obtained from the strain measurements.


## 3.2.2. Mooring Cable Tension Force Estimation

The tension force of the mooring cable is estimated from both the horizontal and vertical displacements at the connection point between the tunnel and mooring cable. For the SFT shown in Figure6, the following equation can be derived from the geometric relationship between the initial t=0 and current t=k∆t states of the SFT:(11)y0+uky−R2+z0+ukz2=Lkl;y0+uky+R−Dy/22+z0+ukz2=Lkr,where R denotes the outer radius of the tunnel and Dy denotes the distance between two anchoring points along the y-direction. y0,z0 denotes the initial position of the center of the tunnel cross-section. Lkl and Lkr denote the lengths of the left and right cables at the kth time step, respectively. Lkl and Lkr are related to the initial lengths L0l and L0r, axial stiffness (K), and tension forces TkL and Tkr of the two cables, as follows:(12)Lkl=1+TklKL0l;Lkr=1+TkrKL0r.Figure 6
Relationship between the tunnel displacement and the anchoring positions of two mooring cables.By combining equations (11) and (12), the tension forces can be estimated as follows:(13)TkL=y0+uky−R2+z0+ukz2−L0lKL0l,Tkr=y0+uky+R−Dy/22+z0+ukz2−L0rKL0r.Note that the mooring cables are assumed to be straight without any sag and are always in tension because of the large initial tension.


## 4. Numerical Validation

### 4.1. Model Description

Figure7(a) shows the model for an SFT, which consists of an 800 m-long tunnel and 54 m-long mooring cables spaced every 50 m along the longitudinal direction of the tunnel. The tunnel was modeled in ABAQUS with the B31 beam element, and the mooring cables were modeled with the T3D2 truss element, similar to existing studies [31, 32]. The simply supported boundary condition, which has been widely used in previous studies [4, 31, 32], was adopted for the tunnel. The different boundary conditions are investigated in Section 4.4.5. Strain gauges were installed at seven mid-span locations (i.e., 5, 9, 13, 15, 19, 23, and 27), and accelerometers were installed at locations 1–31, where the displacement needed to be estimated, as shown in Figure 7(a). Note that these thirty-one locations are either the connection points between the cable and tunnel or the mid-span locations. Additional parameters regarding the tunnel and mooring cables are shown in Figure 7(b), and the values reported in previous studies [31, 33] are used here. The gravity and buoyancy of the tunnel were 1.065 × 106 kN and 2.199 × 106 kN, respectively. The remaining buoyancy of 1.134 × 106 kN was balanced by 30 mooring cables, resulting in 3.78 × 104 kN initial tension force for each mooring cable.Figure 7
Numerical simulation of an SFT model in ABAQUS: (a) overview of the model and sensor setup, (b) design parameters of the model, and (c) cross-sectional view of the model and hydrodynamic environment considered for the model.As shown in Figure7(c), strain gauges were installed at four points on each cross-section at locations 5, 9, 13, 15, 19, 23, and 27, for horizontal and vertical displacement estimations. Note that for the B31 beam element, only the strain responses at these four points are available in ABAQUS. The tunnel had a tubular cross-section with an inner diameter of 18 m and a thickness of 1 m. One end of the mooring cable, inclined at 45°, was pin-connected to the tunnel and the other end to the seabed. The distances from the tunnel to the seabed and the water surface were set to 38.5 m and 61.5 m, respectively, similar to the previous studies [33–35]. The SFT model was simultaneously subjected to (1) wave and current loadings and (2) uniformly distributed loading with random amplitudes in both the horizontal and vertical directions, as shown in Figure 7(c). The hydrodynamic environment was simulated using ABAQUS/Aqua, which has been commonly used in previous studies [36–38]. However, the real hydrodynamic environment may be more complex [39], and more research is needed to consider this.Then, the corresponding acceleration, strain, displacement responses, and cable tension forces were simulated for 100 s at a sampling rate of 100 Hz. The simulated displacement and cable tension force were used as reference values to evaluate the estimation performance of the proposed technique. The current velocity decreased linearly from 0.1 m/s at the water surface to 0 at the seabed. The wave height and period were 11.7 m and 13 s, respectively. The hydrodynamic environment was simulated using ABAQUS/Aqua.


### 4.2. Mode-Scaling Factor Estimation Results

Strain gauges were installed at seven locations, and seven modes were extracted. Mode-scaling factors were estimated from the first three modes (P=3) using the 30 s acceleration responses recorded at three locations (i.e., 5, 15, and 27) and strain responses at all seven locations. Because the first resonance frequency of the SFT was 0.35 Hz, the cutoff frequency of the high-pass filter required for the mode-scaling factor estimation was set to 0.30 Hz. As shown in Figure 8(a), three different modal responses at 0.35 Hz, 0.49 Hz, and 0.96 Hz were decomposed from strain and acceleration measurements using equations (7) and (9), respectively.Figure 8
(a) Decomposed modal responses using equations (7) and (9) and (b) estimated mode-scaling factors based on equation (10) using the first 30 s acceleration responses at three locations (i.e., 5, 15, and 27) and strain responses at seven locations (i.e., 5, 9, 13, 15, 19, 23, and 27).
(a)(b)Here, only the first three modes were used for the displacement estimation because the contribution of the lower modes is more significant than that of the higher modes. That is, as the mode order increases, the magnitude of the modal responses decreases significantly. For example, the response amplitude of the third-order mode was only 1.4% of the first-order mode response. The corresponding scale-factor estimation results are shown in Figure8(b). Apparent linear relationships were observed between the acceleration- and strain-based modal responses, with R2 values greater than 0.97. Note that all results in this section were obtained in the vertical direction, and similar results obtained in the horizontal direction were omitted here owing to space limitations.Figure9 shows a comparison of the first three true strain mode shapes of the SFT model and the corresponding simplified strain mode shapes. As mentioned in Section 2.1, the discrepancy decreased from the first-mode shape to the third mode shape, and the difference was negligible for the third mode. Even for the first-mode shape, good agreement was observed at mid-span locations (i.e., locations 5, 9, 13, 15, 19, 23, and 27) after scaling of the simplified first mode. Therefore, the use of simplified strain mode shapes is acceptable if the strain responses at mid-span locations are used. For the SFT model used in this simulation, the stiffness ratio defined in equation (1) was 0.0122, which is close to the values used in Jin et al. [33] and Kim et al. [4] (0.0125 and 0.013). The effect of the stiffness ratio on the displacement estimation accuracy is discussed in Section 4.4.4.Figure 9
Comparison of true strain mode shapes and simplified mode shapes of the SFT model after amplitude scaling.


### 4.3. Displacement and Tension Force Estimation Results

#### 4.3.1. Displacements

The cutoff frequency of the FIR filters was set to the first resonance frequency of the SFT, i.e., 0.35 Hz. The displacement at location 1 was estimated using the acceleration response at this location and the strain responses at locations 5, 9, 13, 15, 19, 23, and 27. This procedure was repeated for all thirty one locations where accelerometers were placed to estimate the displacements at all locations. Figure10 shows the displacements estimated at the middle of the tunnel (location 15) using the proposed and existing techniques [16]. The existing technique first transforms strain measurements to displacement using the assumed mode shapes and then simultaneously estimates the final displacement and a scaling factor, used to compensate for the discrepancy between the true and assumed mode shapes from the transformed displacement and acceleration measurement by integrating the FIR filter with a recursive least square (RLS) algorithm. The displacements estimated using both techniques agreed closely with the reference displacements. Note that the existing technique computes a single scaling factor for all modes, but the scaling factor value varies with respect to the displacement estimation locations. In contrast, the proposed technique computes different scaling factors for each mode; however, each scaling factor value was constant at all locations. The estimated displacements from 15 to 18 s were magnified on the right-hand side of Figure 10 to clearly show the differences between the estimated and reference displacements.Figure 10
Comparison of (a) vertical and (b) horizontal displacements estimated from the proposed and existing techniques [16] at location 15 of the numerical SFT model.
(a)(b)The displacement estimation performance was quantitatively evaluated using the root mean square error (RMSE) and normalized RMSE (NRMSE) as follows:(14)RMSEmm=∑k=1Qukr−uk2Q,NRMSE%=100RMSEmaxukr−minukr,where ukr and uk denote the reference and estimated displacements at the kth time step, respectively, and Q is the number of data points. The proposed technique estimated displacements with NRMSE and RMSE values of less than 0.5% and 0.7 mm, respectively. Note that the NRMSE and RMSE values of the proposed technique were only one-fifth those of the existing technique.Figure11 summarizes the NRMSE and RMSE values of the estimated horizontal and vertical displacements at all thirty-one locations. The proposed technique had smaller NRMSE and RMSE values than the existing technique at most locations. The NRMSE and RMSE values achieved by the proposed method were below 1% and 1 mm, respectively, in both horizontal and vertical directions. In addition, the displacement estimation accuracy of the proposed technique was less sensitive to the displacement estimation location than that of the existing technique, and a similar level of performance was achieved at all locations. Note that the proposed technique estimated only three location-independent mode-scaling factors, whereas the existing technique required thirty-one location-dependent scaling factors to estimate the displacement at thirty-one locations.Figure 11
Comparison of (a) NRMSE and (b) RMSE of displacements estimated by the proposed and existing techniques at different locations of the numerical SFT model.
(a)(b)


#### 4.3.2. Tension Forces

The displacement estimated at the connection point between the tunnel and mooring cable was used to estimate the tension force of the corresponding mooring cable. Figure12 shows the tension force estimated at the left mooring cable for location 16. Good agreement was observed between the estimated and reference tension forces. Figure 13 summarizes the NRMSE values of the tension forces estimated for all mooring cables. The proposed technique showed less than 2% NRMSE and 60 kN RMSE for all mooring cables.Figure 12
Tension force estimated using the proposed technique for the left mooring cable at location 16 of the numerical SFT model.Figure 13
Comparison of (a) NRMSE and (b) RMSE values of tension forces estimated at all mooring cables of the numerical SFT model.
(a)(b)


### 4.4. Parametric Study on the Displacement Estimation Accuracy

#### 4.4.1. Measurement Noises

Different levels of Gaussian white noise were added to the acceleration and strain responses to examine the effect of measurement noise on the displacement measurement accuracy. Because accelerometers typically have higher signal-to-noise ratios (SNRs) than strain gauges, the SNR of the acceleration responses was set to twice that of the strain responses. Figure14 summarizes the displacement estimation performance under different measurement noise levels. Although the displacement estimation error increased with the measurement noise level, it was still less than 2% at most locations. Note that the NRMSE values were relatively large near the two fixed ends because of the relatively small displacements at these locations, particularly in the horizontal direction. However, the absolute displacement errors at these locations were very small, less than 1.5 mm.Figure 14
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying strain and acceleration measurement noise levels.
(a)(b)


#### 4.4.2. Values ofP and N

First, the displacements were estimated by varying theN value from one to nine. The corresponding RMSEs of the estimated displacements are shown in Figure 15. Significant RMSE reductions were observed when N increased from one to seven. However, further increases beyond seven did not improve the NRMSE value much. The P value was set to one when N equaled one and to three in other cases.Figure 15
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel with varyingN values.
(a)(b)Next, theN value was fixed at seven, and the displacements were estimated by varying the P value. The corresponding NRMSE values of the estimated displacements are shown in Figure 16. Increasing the P value from one to two resulted in a significant decrease in the NRMSE value. However, further increase of the P value did not reduce the NRMSE value much. This observation is consistent with Figure 8(a). As the mode order increases, the magnitude of the modal responses decreases significantly. Therefore, setting the values of the high-order scaling factors to one did not cause a significant difference.Figure 16
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel with varyingP values.
(a)(b)


#### 4.4.3. Wave Period

The wave period was changed from 13 s to 3 s, 7 s, 25 s, and 40 s, and the corresponding displacements were estimated using the proposed technique. It should be noted that the SFT model at low frequencies is mainly excited by wave loading. The SFT model had low-frequency vibration at different frequencies when the wave period was changed. Figure17 summarizes the displacement estimation accuracies for different wave periods. Although slight differences were observed, the proposed technique estimated displacement accurately with less than 1% NRMSE in all cases.Figure 17
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying wave period.
(a)(b)


#### 4.4.4. Stiffness Ratio

Three additional SFT models with different stiffness ratios were simulated by changing the stiffness of the mooring cables. Similar effects can be obtained by changing the distance between two adjacent mooring cables or the bending stiffness of the tunnel. Note that the stiffness ratios of all models still satisfy equation (1). The displacements were estimated for the four SFT models following the procedures described in Section 4.1, except for the stiffness ratios. Figure 18 compares the true and simplified first-mode shapes of the four SFT models. Apparently, the discrepancy increases as the stiffness ratio increases, resulting in larger mode-scaling factors. For all models, the proposed technique estimated displacement with less than 2% NRMSE, as shown in Figure 19.Figure 18
Comparison of the true and simplified first-mode shapes of the SFT models under varying stiffness ratios.Figure 19
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying stiffness ratios.
(a)(b)


#### 4.4.5. Boundary Condition

By changing the boundary conditions of the tunnel from hinged at both ends to fixed at both ends, and fixed at one end and hinged at the other, two additional SFT models were simulated. Note that the real-world construction of an SFT has not yet been realized, and most existing studies have adopted either simply supported [4, 31, 32] or fixed-fixed [3, 33, 35, 40] boundary conditions. Figure 20 summarizes the accuracy of displacement estimation. Because the simplified mode shapes were obtained with simply supported boundary conditions, large discrepancies between the simplified and true mode shapes appeared at locations near the two ends, i.e., locations 1–3 and 29–31 for the SFT model with fixed boundaries at both ends. Therefore, large NRMSE and RMSE values were observed at these six locations (1–3 and 29–31). For the remaining 25 locations, NRMSE values obtained using the proposed technique were below 0.8%. When the SFT model was fixed at one end and hinged at the other, extremely large errors were observed near the fixed end, i.e., at locations 1–3. Except for these three locations, displacements were still reliably estimated at the remaining 28 locations, with RMSE values less than 2.4%. Much larger errors were observed from the SFT model fixed at one end and hinged at the other owing to its nonsymmetric mode shapes.Figure 20
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying boundary conditions.
(a)(b)


## 4.1. Model Description

Figure7(a) shows the model for an SFT, which consists of an 800 m-long tunnel and 54 m-long mooring cables spaced every 50 m along the longitudinal direction of the tunnel. The tunnel was modeled in ABAQUS with the B31 beam element, and the mooring cables were modeled with the T3D2 truss element, similar to existing studies [31, 32]. The simply supported boundary condition, which has been widely used in previous studies [4, 31, 32], was adopted for the tunnel. The different boundary conditions are investigated in Section 4.4.5. Strain gauges were installed at seven mid-span locations (i.e., 5, 9, 13, 15, 19, 23, and 27), and accelerometers were installed at locations 1–31, where the displacement needed to be estimated, as shown in Figure 7(a). Note that these thirty-one locations are either the connection points between the cable and tunnel or the mid-span locations. Additional parameters regarding the tunnel and mooring cables are shown in Figure 7(b), and the values reported in previous studies [31, 33] are used here. The gravity and buoyancy of the tunnel were 1.065 × 106 kN and 2.199 × 106 kN, respectively. The remaining buoyancy of 1.134 × 106 kN was balanced by 30 mooring cables, resulting in 3.78 × 104 kN initial tension force for each mooring cable.Figure 7
Numerical simulation of an SFT model in ABAQUS: (a) overview of the model and sensor setup, (b) design parameters of the model, and (c) cross-sectional view of the model and hydrodynamic environment considered for the model.As shown in Figure7(c), strain gauges were installed at four points on each cross-section at locations 5, 9, 13, 15, 19, 23, and 27, for horizontal and vertical displacement estimations. Note that for the B31 beam element, only the strain responses at these four points are available in ABAQUS. The tunnel had a tubular cross-section with an inner diameter of 18 m and a thickness of 1 m. One end of the mooring cable, inclined at 45°, was pin-connected to the tunnel and the other end to the seabed. The distances from the tunnel to the seabed and the water surface were set to 38.5 m and 61.5 m, respectively, similar to the previous studies [33–35]. The SFT model was simultaneously subjected to (1) wave and current loadings and (2) uniformly distributed loading with random amplitudes in both the horizontal and vertical directions, as shown in Figure 7(c). The hydrodynamic environment was simulated using ABAQUS/Aqua, which has been commonly used in previous studies [36–38]. However, the real hydrodynamic environment may be more complex [39], and more research is needed to consider this.Then, the corresponding acceleration, strain, displacement responses, and cable tension forces were simulated for 100 s at a sampling rate of 100 Hz. The simulated displacement and cable tension force were used as reference values to evaluate the estimation performance of the proposed technique. The current velocity decreased linearly from 0.1 m/s at the water surface to 0 at the seabed. The wave height and period were 11.7 m and 13 s, respectively. The hydrodynamic environment was simulated using ABAQUS/Aqua.


## 4.2. Mode-Scaling Factor Estimation Results

Strain gauges were installed at seven locations, and seven modes were extracted. Mode-scaling factors were estimated from the first three modes (P=3) using the 30 s acceleration responses recorded at three locations (i.e., 5, 15, and 27) and strain responses at all seven locations. Because the first resonance frequency of the SFT was 0.35 Hz, the cutoff frequency of the high-pass filter required for the mode-scaling factor estimation was set to 0.30 Hz. As shown in Figure 8(a), three different modal responses at 0.35 Hz, 0.49 Hz, and 0.96 Hz were decomposed from strain and acceleration measurements using equations (7) and (9), respectively.Figure 8
(a) Decomposed modal responses using equations (7) and (9) and (b) estimated mode-scaling factors based on equation (10) using the first 30 s acceleration responses at three locations (i.e., 5, 15, and 27) and strain responses at seven locations (i.e., 5, 9, 13, 15, 19, 23, and 27).
(a)(b)Here, only the first three modes were used for the displacement estimation because the contribution of the lower modes is more significant than that of the higher modes. That is, as the mode order increases, the magnitude of the modal responses decreases significantly. For example, the response amplitude of the third-order mode was only 1.4% of the first-order mode response. The corresponding scale-factor estimation results are shown in Figure8(b). Apparent linear relationships were observed between the acceleration- and strain-based modal responses, with R2 values greater than 0.97. Note that all results in this section were obtained in the vertical direction, and similar results obtained in the horizontal direction were omitted here owing to space limitations.Figure9 shows a comparison of the first three true strain mode shapes of the SFT model and the corresponding simplified strain mode shapes. As mentioned in Section 2.1, the discrepancy decreased from the first-mode shape to the third mode shape, and the difference was negligible for the third mode. Even for the first-mode shape, good agreement was observed at mid-span locations (i.e., locations 5, 9, 13, 15, 19, 23, and 27) after scaling of the simplified first mode. Therefore, the use of simplified strain mode shapes is acceptable if the strain responses at mid-span locations are used. For the SFT model used in this simulation, the stiffness ratio defined in equation (1) was 0.0122, which is close to the values used in Jin et al. [33] and Kim et al. [4] (0.0125 and 0.013). The effect of the stiffness ratio on the displacement estimation accuracy is discussed in Section 4.4.4.Figure 9
Comparison of true strain mode shapes and simplified mode shapes of the SFT model after amplitude scaling.


## 4.3. Displacement and Tension Force Estimation Results

### 4.3.1. Displacements

The cutoff frequency of the FIR filters was set to the first resonance frequency of the SFT, i.e., 0.35 Hz. The displacement at location 1 was estimated using the acceleration response at this location and the strain responses at locations 5, 9, 13, 15, 19, 23, and 27. This procedure was repeated for all thirty one locations where accelerometers were placed to estimate the displacements at all locations. Figure10 shows the displacements estimated at the middle of the tunnel (location 15) using the proposed and existing techniques [16]. The existing technique first transforms strain measurements to displacement using the assumed mode shapes and then simultaneously estimates the final displacement and a scaling factor, used to compensate for the discrepancy between the true and assumed mode shapes from the transformed displacement and acceleration measurement by integrating the FIR filter with a recursive least square (RLS) algorithm. The displacements estimated using both techniques agreed closely with the reference displacements. Note that the existing technique computes a single scaling factor for all modes, but the scaling factor value varies with respect to the displacement estimation locations. In contrast, the proposed technique computes different scaling factors for each mode; however, each scaling factor value was constant at all locations. The estimated displacements from 15 to 18 s were magnified on the right-hand side of Figure 10 to clearly show the differences between the estimated and reference displacements.Figure 10
Comparison of (a) vertical and (b) horizontal displacements estimated from the proposed and existing techniques [16] at location 15 of the numerical SFT model.
(a)(b)The displacement estimation performance was quantitatively evaluated using the root mean square error (RMSE) and normalized RMSE (NRMSE) as follows:(14)RMSEmm=∑k=1Qukr−uk2Q,NRMSE%=100RMSEmaxukr−minukr,where ukr and uk denote the reference and estimated displacements at the kth time step, respectively, and Q is the number of data points. The proposed technique estimated displacements with NRMSE and RMSE values of less than 0.5% and 0.7 mm, respectively. Note that the NRMSE and RMSE values of the proposed technique were only one-fifth those of the existing technique.Figure11 summarizes the NRMSE and RMSE values of the estimated horizontal and vertical displacements at all thirty-one locations. The proposed technique had smaller NRMSE and RMSE values than the existing technique at most locations. The NRMSE and RMSE values achieved by the proposed method were below 1% and 1 mm, respectively, in both horizontal and vertical directions. In addition, the displacement estimation accuracy of the proposed technique was less sensitive to the displacement estimation location than that of the existing technique, and a similar level of performance was achieved at all locations. Note that the proposed technique estimated only three location-independent mode-scaling factors, whereas the existing technique required thirty-one location-dependent scaling factors to estimate the displacement at thirty-one locations.Figure 11
Comparison of (a) NRMSE and (b) RMSE of displacements estimated by the proposed and existing techniques at different locations of the numerical SFT model.
(a)(b)


### 4.3.2. Tension Forces

The displacement estimated at the connection point between the tunnel and mooring cable was used to estimate the tension force of the corresponding mooring cable. Figure12 shows the tension force estimated at the left mooring cable for location 16. Good agreement was observed between the estimated and reference tension forces. Figure 13 summarizes the NRMSE values of the tension forces estimated for all mooring cables. The proposed technique showed less than 2% NRMSE and 60 kN RMSE for all mooring cables.Figure 12
Tension force estimated using the proposed technique for the left mooring cable at location 16 of the numerical SFT model.Figure 13
Comparison of (a) NRMSE and (b) RMSE values of tension forces estimated at all mooring cables of the numerical SFT model.
(a)(b)


## 4.3.1. Displacements

The cutoff frequency of the FIR filters was set to the first resonance frequency of the SFT, i.e., 0.35 Hz. The displacement at location 1 was estimated using the acceleration response at this location and the strain responses at locations 5, 9, 13, 15, 19, 23, and 27. This procedure was repeated for all thirty one locations where accelerometers were placed to estimate the displacements at all locations. Figure10 shows the displacements estimated at the middle of the tunnel (location 15) using the proposed and existing techniques [16]. The existing technique first transforms strain measurements to displacement using the assumed mode shapes and then simultaneously estimates the final displacement and a scaling factor, used to compensate for the discrepancy between the true and assumed mode shapes from the transformed displacement and acceleration measurement by integrating the FIR filter with a recursive least square (RLS) algorithm. The displacements estimated using both techniques agreed closely with the reference displacements. Note that the existing technique computes a single scaling factor for all modes, but the scaling factor value varies with respect to the displacement estimation locations. In contrast, the proposed technique computes different scaling factors for each mode; however, each scaling factor value was constant at all locations. The estimated displacements from 15 to 18 s were magnified on the right-hand side of Figure 10 to clearly show the differences between the estimated and reference displacements.Figure 10
Comparison of (a) vertical and (b) horizontal displacements estimated from the proposed and existing techniques [16] at location 15 of the numerical SFT model.
(a)(b)The displacement estimation performance was quantitatively evaluated using the root mean square error (RMSE) and normalized RMSE (NRMSE) as follows:(14)RMSEmm=∑k=1Qukr−uk2Q,NRMSE%=100RMSEmaxukr−minukr,where ukr and uk denote the reference and estimated displacements at the kth time step, respectively, and Q is the number of data points. The proposed technique estimated displacements with NRMSE and RMSE values of less than 0.5% and 0.7 mm, respectively. Note that the NRMSE and RMSE values of the proposed technique were only one-fifth those of the existing technique.Figure11 summarizes the NRMSE and RMSE values of the estimated horizontal and vertical displacements at all thirty-one locations. The proposed technique had smaller NRMSE and RMSE values than the existing technique at most locations. The NRMSE and RMSE values achieved by the proposed method were below 1% and 1 mm, respectively, in both horizontal and vertical directions. In addition, the displacement estimation accuracy of the proposed technique was less sensitive to the displacement estimation location than that of the existing technique, and a similar level of performance was achieved at all locations. Note that the proposed technique estimated only three location-independent mode-scaling factors, whereas the existing technique required thirty-one location-dependent scaling factors to estimate the displacement at thirty-one locations.Figure 11
Comparison of (a) NRMSE and (b) RMSE of displacements estimated by the proposed and existing techniques at different locations of the numerical SFT model.
(a)(b)


## 4.3.2. Tension Forces

The displacement estimated at the connection point between the tunnel and mooring cable was used to estimate the tension force of the corresponding mooring cable. Figure12 shows the tension force estimated at the left mooring cable for location 16. Good agreement was observed between the estimated and reference tension forces. Figure 13 summarizes the NRMSE values of the tension forces estimated for all mooring cables. The proposed technique showed less than 2% NRMSE and 60 kN RMSE for all mooring cables.Figure 12
Tension force estimated using the proposed technique for the left mooring cable at location 16 of the numerical SFT model.Figure 13
Comparison of (a) NRMSE and (b) RMSE values of tension forces estimated at all mooring cables of the numerical SFT model.
(a)(b)


## 4.4. Parametric Study on the Displacement Estimation Accuracy

### 4.4.1. Measurement Noises

Different levels of Gaussian white noise were added to the acceleration and strain responses to examine the effect of measurement noise on the displacement measurement accuracy. Because accelerometers typically have higher signal-to-noise ratios (SNRs) than strain gauges, the SNR of the acceleration responses was set to twice that of the strain responses. Figure14 summarizes the displacement estimation performance under different measurement noise levels. Although the displacement estimation error increased with the measurement noise level, it was still less than 2% at most locations. Note that the NRMSE values were relatively large near the two fixed ends because of the relatively small displacements at these locations, particularly in the horizontal direction. However, the absolute displacement errors at these locations were very small, less than 1.5 mm.Figure 14
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying strain and acceleration measurement noise levels.
(a)(b)


### 4.4.2. Values ofP and N

First, the displacements were estimated by varying theN value from one to nine. The corresponding RMSEs of the estimated displacements are shown in Figure 15. Significant RMSE reductions were observed when N increased from one to seven. However, further increases beyond seven did not improve the NRMSE value much. The P value was set to one when N equaled one and to three in other cases.Figure 15
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel with varyingN values.
(a)(b)Next, theN value was fixed at seven, and the displacements were estimated by varying the P value. The corresponding NRMSE values of the estimated displacements are shown in Figure 16. Increasing the P value from one to two resulted in a significant decrease in the NRMSE value. However, further increase of the P value did not reduce the NRMSE value much. This observation is consistent with Figure 8(a). As the mode order increases, the magnitude of the modal responses decreases significantly. Therefore, setting the values of the high-order scaling factors to one did not cause a significant difference.Figure 16
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel with varyingP values.
(a)(b)


### 4.4.3. Wave Period

The wave period was changed from 13 s to 3 s, 7 s, 25 s, and 40 s, and the corresponding displacements were estimated using the proposed technique. It should be noted that the SFT model at low frequencies is mainly excited by wave loading. The SFT model had low-frequency vibration at different frequencies when the wave period was changed. Figure17 summarizes the displacement estimation accuracies for different wave periods. Although slight differences were observed, the proposed technique estimated displacement accurately with less than 1% NRMSE in all cases.Figure 17
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying wave period.
(a)(b)


### 4.4.4. Stiffness Ratio

Three additional SFT models with different stiffness ratios were simulated by changing the stiffness of the mooring cables. Similar effects can be obtained by changing the distance between two adjacent mooring cables or the bending stiffness of the tunnel. Note that the stiffness ratios of all models still satisfy equation (1). The displacements were estimated for the four SFT models following the procedures described in Section 4.1, except for the stiffness ratios. Figure 18 compares the true and simplified first-mode shapes of the four SFT models. Apparently, the discrepancy increases as the stiffness ratio increases, resulting in larger mode-scaling factors. For all models, the proposed technique estimated displacement with less than 2% NRMSE, as shown in Figure 19.Figure 18
Comparison of the true and simplified first-mode shapes of the SFT models under varying stiffness ratios.Figure 19
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying stiffness ratios.
(a)(b)


### 4.4.5. Boundary Condition

By changing the boundary conditions of the tunnel from hinged at both ends to fixed at both ends, and fixed at one end and hinged at the other, two additional SFT models were simulated. Note that the real-world construction of an SFT has not yet been realized, and most existing studies have adopted either simply supported [4, 31, 32] or fixed-fixed [3, 33, 35, 40] boundary conditions. Figure 20 summarizes the accuracy of displacement estimation. Because the simplified mode shapes were obtained with simply supported boundary conditions, large discrepancies between the simplified and true mode shapes appeared at locations near the two ends, i.e., locations 1–3 and 29–31 for the SFT model with fixed boundaries at both ends. Therefore, large NRMSE and RMSE values were observed at these six locations (1–3 and 29–31). For the remaining 25 locations, NRMSE values obtained using the proposed technique were below 0.8%. When the SFT model was fixed at one end and hinged at the other, extremely large errors were observed near the fixed end, i.e., at locations 1–3. Except for these three locations, displacements were still reliably estimated at the remaining 28 locations, with RMSE values less than 2.4%. Much larger errors were observed from the SFT model fixed at one end and hinged at the other owing to its nonsymmetric mode shapes.Figure 20
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying boundary conditions.
(a)(b)


## 4.4.1. Measurement Noises

Different levels of Gaussian white noise were added to the acceleration and strain responses to examine the effect of measurement noise on the displacement measurement accuracy. Because accelerometers typically have higher signal-to-noise ratios (SNRs) than strain gauges, the SNR of the acceleration responses was set to twice that of the strain responses. Figure14 summarizes the displacement estimation performance under different measurement noise levels. Although the displacement estimation error increased with the measurement noise level, it was still less than 2% at most locations. Note that the NRMSE values were relatively large near the two fixed ends because of the relatively small displacements at these locations, particularly in the horizontal direction. However, the absolute displacement errors at these locations were very small, less than 1.5 mm.Figure 14
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying strain and acceleration measurement noise levels.
(a)(b)


## 4.4.2. Values ofP and N

First, the displacements were estimated by varying theN value from one to nine. The corresponding RMSEs of the estimated displacements are shown in Figure 15. Significant RMSE reductions were observed when N increased from one to seven. However, further increases beyond seven did not improve the NRMSE value much. The P value was set to one when N equaled one and to three in other cases.Figure 15
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel with varyingN values.
(a)(b)Next, theN value was fixed at seven, and the displacements were estimated by varying the P value. The corresponding NRMSE values of the estimated displacements are shown in Figure 16. Increasing the P value from one to two resulted in a significant decrease in the NRMSE value. However, further increase of the P value did not reduce the NRMSE value much. This observation is consistent with Figure 8(a). As the mode order increases, the magnitude of the modal responses decreases significantly. Therefore, setting the values of the high-order scaling factors to one did not cause a significant difference.Figure 16
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel with varyingP values.
(a)(b)


## 4.4.3. Wave Period

The wave period was changed from 13 s to 3 s, 7 s, 25 s, and 40 s, and the corresponding displacements were estimated using the proposed technique. It should be noted that the SFT model at low frequencies is mainly excited by wave loading. The SFT model had low-frequency vibration at different frequencies when the wave period was changed. Figure17 summarizes the displacement estimation accuracies for different wave periods. Although slight differences were observed, the proposed technique estimated displacement accurately with less than 1% NRMSE in all cases.Figure 17
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying wave period.
(a)(b)


## 4.4.4. Stiffness Ratio

Three additional SFT models with different stiffness ratios were simulated by changing the stiffness of the mooring cables. Similar effects can be obtained by changing the distance between two adjacent mooring cables or the bending stiffness of the tunnel. Note that the stiffness ratios of all models still satisfy equation (1). The displacements were estimated for the four SFT models following the procedures described in Section 4.1, except for the stiffness ratios. Figure 18 compares the true and simplified first-mode shapes of the four SFT models. Apparently, the discrepancy increases as the stiffness ratio increases, resulting in larger mode-scaling factors. For all models, the proposed technique estimated displacement with less than 2% NRMSE, as shown in Figure 19.Figure 18
Comparison of the true and simplified first-mode shapes of the SFT models under varying stiffness ratios.Figure 19
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying stiffness ratios.
(a)(b)


## 4.4.5. Boundary Condition

By changing the boundary conditions of the tunnel from hinged at both ends to fixed at both ends, and fixed at one end and hinged at the other, two additional SFT models were simulated. Note that the real-world construction of an SFT has not yet been realized, and most existing studies have adopted either simply supported [4, 31, 32] or fixed-fixed [3, 33, 35, 40] boundary conditions. Figure 20 summarizes the accuracy of displacement estimation. Because the simplified mode shapes were obtained with simply supported boundary conditions, large discrepancies between the simplified and true mode shapes appeared at locations near the two ends, i.e., locations 1–3 and 29–31 for the SFT model with fixed boundaries at both ends. Therefore, large NRMSE and RMSE values were observed at these six locations (1–3 and 29–31). For the remaining 25 locations, NRMSE values obtained using the proposed technique were below 0.8%. When the SFT model was fixed at one end and hinged at the other, extremely large errors were observed near the fixed end, i.e., at locations 1–3. Except for these three locations, displacements were still reliably estimated at the remaining 28 locations, with RMSE values less than 2.4%. Much larger errors were observed from the SFT model fixed at one end and hinged at the other owing to its nonsymmetric mode shapes.Figure 20
Accuracy of (a) vertical and (b) horizontal displacements estimated along the longitudinal direction of the tunnel under varying boundary conditions.
(a)(b)


## 5. Experimental Validation

### 5.1. Experimental Setup

To further validate the performance of the proposed technique, a laboratory test was conducted on an 8 m-long SFT mock-up structure. The tunnel consisted of eight bolt-connected aluminum tubes. Note that the Young’s modulus and density of the tube were 70 GPa and 2710 kg/m3, respectively. Each tube was 1 m long and had a uniform circular cross-section with an inner diameter of 13 cm and a thickness of 0.5 cm. The tunnel was submerged into a 12 m long, 10 m wide, and 2 m deep ocean basin, as shown in Figure 21(a), and the tunnel was supported at its ends by a steel frame to simulate a hinge connection, as shown in Figure 21(b).Figure 21
Configuration of the 8 m-long aluminum SFT mock-up test: (a) an SFT mock-up structure installed at a 3D ocean basin, (b) overview of the SFT mock-up structure and sensor locations, (c) a triaxial MEMS accelerometer installed at each of the locations 3–7, and (d) three resistance-type strain gauges installed at each cross-section of locations 1, 2, 4, 6, 8, and 9.Stainless steel wires of 0.5 mm diameter were used as mooring cables, and a total of 14 mooring cables were spaced equally at 1 m intervals along the longitudinal direction of the tunnel, as shown in Figure21(b). For each mooring cable, one end was connected to the tunnel using metal ring buckles and the other end was connected to the pad eye welded to a steel frame. The steel frame was fixed at the bottom of the 3D ocean basin using anchor bolts.Five triaxial MEMS accelerometers, shown in Figure21(c), were attached at locations 3–7, and 18 resistance-type strain gauges, shown in Figure 21(d), were attached at locations 1, 2, 4, 6, 8, and 9. Three strain gauges were installed at each cross-section at locations 1, 2, 4, 6, 8, and 9 for bidirectional displacement estimation. Note that all strain gauges and accelerometers were installed on the external surface of the SFT mock-up structure and waterproofed. Fourteen load cells were adopted for all the mooring cables to measure the reference tension forces. All acceleration, strain, and load cell data were sampled at 100 Hz using National Instruments data acquisition devices. Table 1 lists the detailed specifications of these sensors.Table 1
Detailed specifications of accelerometer, strain gauge, and load cells used in the 8 m-long SFT mock-up test.
SensorsParametersValuesAccelerometers (PCB Piezotronics 3713B112G)BandwidthDC—250 HzFull-scale range±2 gOperating temperature−54∼121°CSensitivity1,000 mV/gSpectral noise22.9μg/HzLinearity<1%Strain gauges (TML FLA-5-11-1LJC)Grid materialCu-Ni alloyGauge factor2.10±1%Gauge resistance120±0.5 ΩOperating temperature−20∼80°CTransverse sensitivity−0.5%Load cells (CASSCALE SMNT-13b)Nonlinearity1%Zero-balance±2%Repeatability1%Input/output impedance350±3 ΩOperating temperature−20∼80°CUltimate load limit200%After sensor installation, water was poured into the 3D ocean basin to ensure that the SFT mock-up structure was immersed and floating. By adjusting the length of the mooring cables, the initial tension forces were set to approximately 150 N. Attempts were made to excite the test structure using different wave loadings, but no meaningful vibration was observed owing to the relatively high stiffness of the SFT mock-up structure. Subsequently, the SFT mock-up structure was excited by manually pushing and releasing the tunnel along three different directions: (1) horizontal (y), (2) vertical (z), and (3) inclined, to generate millimeter-level displacements. Under inclined excitation, displacement was generated in both horizontal and vertical directions. Special attention was paid to minimizing the displacement in the longitudinal (x) direction. Owing to the limited data acquisition channels, only the load cells of cables 1, 2, and 3 (as shown in Figure 21(b)) were used to measure the reference tension forces during the three excitations, whereas the other 11 load cells were used only to adjust the initial tension forces.


### 5.2. Reference Displacement Measurement Using Computer Vision

Computer vision was adopted in this experiment to measure the reference (ground-truth) displacement. Because the tunnel was submerged during the experiment, it was difficult to directly measure its displacement. Instead, five 20 cm-long rigid steel bars with two artificial markers, as shown in Figure22, were attached vertically to the top surface of the tunnel at locations 3–7. Because the tunnel may be subjected to torsion along the x axis, two markers were necessary. Five digital cameras (Sony DSC-WX800) were installed at stationary locations to individually track the movement of the markers at each location. Table 2 presents the detailed specifications of the camera. In this test, each camera recorded the movement of two markers at each location with a 1920 × 1080 resolution and 119.88 Hz sampling rate. The distance between the cameras and markers varied from 4 m to 7 m. The water level was always below these markers during the experiments to ensure successful tracking of the markers. A commonly used template-matching algorithm [8] was adopted for marker tracking. Because the original displacement estimated from the vision measurement is in pixel units, a scale factor is required to convert the pixel unit to a length unit. By adjusting the focal length of each camera, a scale factor of 0.1 mm/pixel was achieved for all targets.Figure 22
Vision-based reference displacement measurement using two artificial markers at different heights of the rigid steel bar attached to the tunnel.Table 2
Detailed specifications of digital camera (Sony DSC-WX800).
ParametersValuesSensor type1/2.3 type CMOS sensorNumber of pixelsApproximately 18.2 megapixelsLens typeZeiss Vario-Sonnar lensFocal length4.25∼118 mmOptical zoom28xResolution and sampling rate for video recording1080 P (1920× 1080) at up to 120 Hz4 K (3840× 2160) at up to 30 HzThe reference displacement was computed as the weighted average of the displacements of the two markers, as shown in Figure22.(15)uy=L1L1−L2u1y−L2L1−L2u2y;uz=u1z+u2z2,where L1 and L2 denote the vertical distances between each marker and tunnel cross-section center, respectively. u1y and u1z denote the horizontal and vertical displacements of Marker 1, respectively. u2y and u2z are defined in a similar manner. Because the strain and acceleration measurements were sampled at 100 Hz, the vision-based displacements, which were originally sampled at 119.88 Hz, were downsampled to 100 Hz.


### 5.3. Displacement and Tension Force Estimation Results

#### 5.3.1. Displacements

Figure23 shows the frequency spectra of the acceleration measured at location 3 under an inclined excitation. The contribution of the first mode to the structural displacement was dominant in both the horizontal and vertical directions. Hence, the mode-scaling factors were estimated only for the first mode (P=1) in both directions. The first resonance frequencies of the SFT mock-up structure were observed at 5.16 Hz and 8.02 Hz in the vertical and horizontal directions, respectively.Figure 23
Frequency spectra of (a) vertical and (b) horizontal accelerations measured under the inclined excitation at location 3 of the 8 m-long SFT mock-up structure.
(a)(b)The displacements were estimated at locations 3–7. Figure24 shows the bidirectional displacements estimated at location 3 under an inclined excitation. Here, displacements were estimated from (1) strain measurement only, (2) the combination of strain and acceleration measurements using the existing technique, and (3) the combination of strain and acceleration measurements using the proposed technique. Significantly larger errors were observed for the strain-based displacements, demonstrating the necessity of combining strain and acceleration measurements. The proposed technique estimated displacements with 2.45% and 2.10% NRMSEs in vertical and horizontal directions, respectively. The RMSEs of the estimated displacements were 0.096 and 0.046 mm in the vertical and horizontal directions, respectively. Compared with the existing technique, approximately 15% error reduction was achieved using the proposed technique, as shown in Figure 24. Note that the proposed technique computes different scaling factors for each mode; however, the scaling factor for a specific mode is constant for different displacement estimation locations. In contrast, the existing technique computes a single scaling factor for all modes, and the scaling factor value varies at each displacement estimation location. However, this did not lead to a significant difference in the displacement estimation in this experiment. Because the contribution of the first mode was significantly more dominant than that of any other mode, only a slight improvement was achieved using the proposed technique.Figure 24
Comparison of (a) vertical and (b) horizontal displacements estimated at location 3 of the aluminum SFT mock-up structure under inclined excitation.
(a)(b)The displacement estimation performance at all five locations is summarized in Table3. The overall RMSE and NRMSE of the strain-based displacement were 0.135 mm and 3.26%, respectively. Note that the estimation of strain-based displacement requires mode-scaling factors. Therefore, accelerometers were still required. Combining the acceleration measurements resulted in a significant improvement in displacement estimation accuracy, and around 27% and 40% reductions were obtained by the existing and proposed techniques, respectively, in both RMSE and NRMSEs.Table 3
Displacement estimation results obtained from 8 m-long SFT mock-up structure test.
Excitations (displacement direction)Estimation locationsStrain onlyExisting technique (strain + acceleration)Proposed technique (strain + acceleration)RMSE (mm)NRMSE (%)RMSE (mm)NRMSE (%)RMSE (mm)NRMSE (%)Vertical excitation (vertical)30.0581.880.0581.860.0471.5040.0912.430.0721.920.0601.6150.0721.710.0621.460.0511.2060.0771.740.0671.520.0561.2770.1202.600.1252.700.1032.22Horizontal excitation (horizontal)30.0792.570.0702.290.0611.9940.0942.600.0932.540.0802.2050.1012.510.1052.600.0872.1560.1343.200.1032.450.0892.1170.1563.810.1172.860.0942.30Inclined excitation (vertical)30.1724.390.1132.890.0962.4540.1983.610.1282.340.1051.9250.2203.630.1362.230.1161.9160.2583.950.1592.420.1352.0670.2754.180.1682.550.1362.06Inclined excitation (horizontal)30.1165.320.0562.550.0462.1040.0922.990.0431.400.0361.1850.1003.080.0611.890.0501.5560.1193.720.0782.420.0652.0270.1705.210.1474.540.1223.76Average0.1353.260.0982.370.0821.98


#### 5.3.2. Tension Forces

The displacements obtained at location 3 were used to estimate the tension force of cable 1, and those from location 5 were used to estimate the tension forces at cables 2 and 3. Figure25 presents the tension force of cable 3 under inclined excitation. A good agreement between the estimated and reference tension forces was observed with 11.20 NRMSE and 3% NRMSE. The tension forces estimated for all the cases are summarized in Figure 26. The proposed technique successfully estimated the tension forces, and RMSE and NRMSE values were less than 12 N and 4.6%, respectively. It should be noted that the load cells installed on cables 1 and 2 leaked after horizontal excitation, resulting in the failure of the reference tension force measurement under inclined excitation. Therefore, the corresponding NRMSE values were not calculated.Figure 25
Comparison of the tension force of cable 3 estimated by the proposed technique and measured by a load cell under inclined excitation in the 8 m-long SFT mock-up structure test.Figure 26
Cable tension force estimation error obtained from the 8 m-long SFT mock-up structure test.
(a)(b)


## 5.1. Experimental Setup

To further validate the performance of the proposed technique, a laboratory test was conducted on an 8 m-long SFT mock-up structure. The tunnel consisted of eight bolt-connected aluminum tubes. Note that the Young’s modulus and density of the tube were 70 GPa and 2710 kg/m3, respectively. Each tube was 1 m long and had a uniform circular cross-section with an inner diameter of 13 cm and a thickness of 0.5 cm. The tunnel was submerged into a 12 m long, 10 m wide, and 2 m deep ocean basin, as shown in Figure 21(a), and the tunnel was supported at its ends by a steel frame to simulate a hinge connection, as shown in Figure 21(b).Figure 21
Configuration of the 8 m-long aluminum SFT mock-up test: (a) an SFT mock-up structure installed at a 3D ocean basin, (b) overview of the SFT mock-up structure and sensor locations, (c) a triaxial MEMS accelerometer installed at each of the locations 3–7, and (d) three resistance-type strain gauges installed at each cross-section of locations 1, 2, 4, 6, 8, and 9.Stainless steel wires of 0.5 mm diameter were used as mooring cables, and a total of 14 mooring cables were spaced equally at 1 m intervals along the longitudinal direction of the tunnel, as shown in Figure21(b). For each mooring cable, one end was connected to the tunnel using metal ring buckles and the other end was connected to the pad eye welded to a steel frame. The steel frame was fixed at the bottom of the 3D ocean basin using anchor bolts.Five triaxial MEMS accelerometers, shown in Figure21(c), were attached at locations 3–7, and 18 resistance-type strain gauges, shown in Figure 21(d), were attached at locations 1, 2, 4, 6, 8, and 9. Three strain gauges were installed at each cross-section at locations 1, 2, 4, 6, 8, and 9 for bidirectional displacement estimation. Note that all strain gauges and accelerometers were installed on the external surface of the SFT mock-up structure and waterproofed. Fourteen load cells were adopted for all the mooring cables to measure the reference tension forces. All acceleration, strain, and load cell data were sampled at 100 Hz using National Instruments data acquisition devices. Table 1 lists the detailed specifications of these sensors.Table 1
Detailed specifications of accelerometer, strain gauge, and load cells used in the 8 m-long SFT mock-up test.
SensorsParametersValuesAccelerometers (PCB Piezotronics 3713B112G)BandwidthDC—250 HzFull-scale range±2 gOperating temperature−54∼121°CSensitivity1,000 mV/gSpectral noise22.9μg/HzLinearity<1%Strain gauges (TML FLA-5-11-1LJC)Grid materialCu-Ni alloyGauge factor2.10±1%Gauge resistance120±0.5 ΩOperating temperature−20∼80°CTransverse sensitivity−0.5%Load cells (CASSCALE SMNT-13b)Nonlinearity1%Zero-balance±2%Repeatability1%Input/output impedance350±3 ΩOperating temperature−20∼80°CUltimate load limit200%After sensor installation, water was poured into the 3D ocean basin to ensure that the SFT mock-up structure was immersed and floating. By adjusting the length of the mooring cables, the initial tension forces were set to approximately 150 N. Attempts were made to excite the test structure using different wave loadings, but no meaningful vibration was observed owing to the relatively high stiffness of the SFT mock-up structure. Subsequently, the SFT mock-up structure was excited by manually pushing and releasing the tunnel along three different directions: (1) horizontal (y), (2) vertical (z), and (3) inclined, to generate millimeter-level displacements. Under inclined excitation, displacement was generated in both horizontal and vertical directions. Special attention was paid to minimizing the displacement in the longitudinal (x) direction. Owing to the limited data acquisition channels, only the load cells of cables 1, 2, and 3 (as shown in Figure 21(b)) were used to measure the reference tension forces during the three excitations, whereas the other 11 load cells were used only to adjust the initial tension forces.


## 5.2. Reference Displacement Measurement Using Computer Vision

Computer vision was adopted in this experiment to measure the reference (ground-truth) displacement. Because the tunnel was submerged during the experiment, it was difficult to directly measure its displacement. Instead, five 20 cm-long rigid steel bars with two artificial markers, as shown in Figure22, were attached vertically to the top surface of the tunnel at locations 3–7. Because the tunnel may be subjected to torsion along the x axis, two markers were necessary. Five digital cameras (Sony DSC-WX800) were installed at stationary locations to individually track the movement of the markers at each location. Table 2 presents the detailed specifications of the camera. In this test, each camera recorded the movement of two markers at each location with a 1920 × 1080 resolution and 119.88 Hz sampling rate. The distance between the cameras and markers varied from 4 m to 7 m. The water level was always below these markers during the experiments to ensure successful tracking of the markers. A commonly used template-matching algorithm [8] was adopted for marker tracking. Because the original displacement estimated from the vision measurement is in pixel units, a scale factor is required to convert the pixel unit to a length unit. By adjusting the focal length of each camera, a scale factor of 0.1 mm/pixel was achieved for all targets.Figure 22
Vision-based reference displacement measurement using two artificial markers at different heights of the rigid steel bar attached to the tunnel.Table 2
Detailed specifications of digital camera (Sony DSC-WX800).
ParametersValuesSensor type1/2.3 type CMOS sensorNumber of pixelsApproximately 18.2 megapixelsLens typeZeiss Vario-Sonnar lensFocal length4.25∼118 mmOptical zoom28xResolution and sampling rate for video recording1080 P (1920× 1080) at up to 120 Hz4 K (3840× 2160) at up to 30 HzThe reference displacement was computed as the weighted average of the displacements of the two markers, as shown in Figure22.(15)uy=L1L1−L2u1y−L2L1−L2u2y;uz=u1z+u2z2,where L1 and L2 denote the vertical distances between each marker and tunnel cross-section center, respectively. u1y and u1z denote the horizontal and vertical displacements of Marker 1, respectively. u2y and u2z are defined in a similar manner. Because the strain and acceleration measurements were sampled at 100 Hz, the vision-based displacements, which were originally sampled at 119.88 Hz, were downsampled to 100 Hz.


## 5.3. Displacement and Tension Force Estimation Results

### 5.3.1. Displacements

Figure23 shows the frequency spectra of the acceleration measured at location 3 under an inclined excitation. The contribution of the first mode to the structural displacement was dominant in both the horizontal and vertical directions. Hence, the mode-scaling factors were estimated only for the first mode (P=1) in both directions. The first resonance frequencies of the SFT mock-up structure were observed at 5.16 Hz and 8.02 Hz in the vertical and horizontal directions, respectively.Figure 23
Frequency spectra of (a) vertical and (b) horizontal accelerations measured under the inclined excitation at location 3 of the 8 m-long SFT mock-up structure.
(a)(b)The displacements were estimated at locations 3–7. Figure24 shows the bidirectional displacements estimated at location 3 under an inclined excitation. Here, displacements were estimated from (1) strain measurement only, (2) the combination of strain and acceleration measurements using the existing technique, and (3) the combination of strain and acceleration measurements using the proposed technique. Significantly larger errors were observed for the strain-based displacements, demonstrating the necessity of combining strain and acceleration measurements. The proposed technique estimated displacements with 2.45% and 2.10% NRMSEs in vertical and horizontal directions, respectively. The RMSEs of the estimated displacements were 0.096 and 0.046 mm in the vertical and horizontal directions, respectively. Compared with the existing technique, approximately 15% error reduction was achieved using the proposed technique, as shown in Figure 24. Note that the proposed technique computes different scaling factors for each mode; however, the scaling factor for a specific mode is constant for different displacement estimation locations. In contrast, the existing technique computes a single scaling factor for all modes, and the scaling factor value varies at each displacement estimation location. However, this did not lead to a significant difference in the displacement estimation in this experiment. Because the contribution of the first mode was significantly more dominant than that of any other mode, only a slight improvement was achieved using the proposed technique.Figure 24
Comparison of (a) vertical and (b) horizontal displacements estimated at location 3 of the aluminum SFT mock-up structure under inclined excitation.
(a)(b)The displacement estimation performance at all five locations is summarized in Table3. The overall RMSE and NRMSE of the strain-based displacement were 0.135 mm and 3.26%, respectively. Note that the estimation of strain-based displacement requires mode-scaling factors. Therefore, accelerometers were still required. Combining the acceleration measurements resulted in a significant improvement in displacement estimation accuracy, and around 27% and 40% reductions were obtained by the existing and proposed techniques, respectively, in both RMSE and NRMSEs.Table 3
Displacement estimation results obtained from 8 m-long SFT mock-up structure test.
Excitations (displacement direction)Estimation locationsStrain onlyExisting technique (strain + acceleration)Proposed technique (strain + acceleration)RMSE (mm)NRMSE (%)RMSE (mm)NRMSE (%)RMSE (mm)NRMSE (%)Vertical excitation (vertical)30.0581.880.0581.860.0471.5040.0912.430.0721.920.0601.6150.0721.710.0621.460.0511.2060.0771.740.0671.520.0561.2770.1202.600.1252.700.1032.22Horizontal excitation (horizontal)30.0792.570.0702.290.0611.9940.0942.600.0932.540.0802.2050.1012.510.1052.600.0872.1560.1343.200.1032.450.0892.1170.1563.810.1172.860.0942.30Inclined excitation (vertical)30.1724.390.1132.890.0962.4540.1983.610.1282.340.1051.9250.2203.630.1362.230.1161.9160.2583.950.1592.420.1352.0670.2754.180.1682.550.1362.06Inclined excitation (horizontal)30.1165.320.0562.550.0462.1040.0922.990.0431.400.0361.1850.1003.080.0611.890.0501.5560.1193.720.0782.420.0652.0270.1705.210.1474.540.1223.76Average0.1353.260.0982.370.0821.98


### 5.3.2. Tension Forces

The displacements obtained at location 3 were used to estimate the tension force of cable 1, and those from location 5 were used to estimate the tension forces at cables 2 and 3. Figure25 presents the tension force of cable 3 under inclined excitation. A good agreement between the estimated and reference tension forces was observed with 11.20 NRMSE and 3% NRMSE. The tension forces estimated for all the cases are summarized in Figure 26. The proposed technique successfully estimated the tension forces, and RMSE and NRMSE values were less than 12 N and 4.6%, respectively. It should be noted that the load cells installed on cables 1 and 2 leaked after horizontal excitation, resulting in the failure of the reference tension force measurement under inclined excitation. Therefore, the corresponding NRMSE values were not calculated.Figure 25
Comparison of the tension force of cable 3 estimated by the proposed technique and measured by a load cell under inclined excitation in the 8 m-long SFT mock-up structure test.Figure 26
Cable tension force estimation error obtained from the 8 m-long SFT mock-up structure test.
(a)(b)


## 5.3.1. Displacements

Figure23 shows the frequency spectra of the acceleration measured at location 3 under an inclined excitation. The contribution of the first mode to the structural displacement was dominant in both the horizontal and vertical directions. Hence, the mode-scaling factors were estimated only for the first mode (P=1) in both directions. The first resonance frequencies of the SFT mock-up structure were observed at 5.16 Hz and 8.02 Hz in the vertical and horizontal directions, respectively.Figure 23
Frequency spectra of (a) vertical and (b) horizontal accelerations measured under the inclined excitation at location 3 of the 8 m-long SFT mock-up structure.
(a)(b)The displacements were estimated at locations 3–7. Figure24 shows the bidirectional displacements estimated at location 3 under an inclined excitation. Here, displacements were estimated from (1) strain measurement only, (2) the combination of strain and acceleration measurements using the existing technique, and (3) the combination of strain and acceleration measurements using the proposed technique. Significantly larger errors were observed for the strain-based displacements, demonstrating the necessity of combining strain and acceleration measurements. The proposed technique estimated displacements with 2.45% and 2.10% NRMSEs in vertical and horizontal directions, respectively. The RMSEs of the estimated displacements were 0.096 and 0.046 mm in the vertical and horizontal directions, respectively. Compared with the existing technique, approximately 15% error reduction was achieved using the proposed technique, as shown in Figure 24. Note that the proposed technique computes different scaling factors for each mode; however, the scaling factor for a specific mode is constant for different displacement estimation locations. In contrast, the existing technique computes a single scaling factor for all modes, and the scaling factor value varies at each displacement estimation location. However, this did not lead to a significant difference in the displacement estimation in this experiment. Because the contribution of the first mode was significantly more dominant than that of any other mode, only a slight improvement was achieved using the proposed technique.Figure 24
Comparison of (a) vertical and (b) horizontal displacements estimated at location 3 of the aluminum SFT mock-up structure under inclined excitation.
(a)(b)The displacement estimation performance at all five locations is summarized in Table3. The overall RMSE and NRMSE of the strain-based displacement were 0.135 mm and 3.26%, respectively. Note that the estimation of strain-based displacement requires mode-scaling factors. Therefore, accelerometers were still required. Combining the acceleration measurements resulted in a significant improvement in displacement estimation accuracy, and around 27% and 40% reductions were obtained by the existing and proposed techniques, respectively, in both RMSE and NRMSEs.Table 3
Displacement estimation results obtained from 8 m-long SFT mock-up structure test.
Excitations (displacement direction)Estimation locationsStrain onlyExisting technique (strain + acceleration)Proposed technique (strain + acceleration)RMSE (mm)NRMSE (%)RMSE (mm)NRMSE (%)RMSE (mm)NRMSE (%)Vertical excitation (vertical)30.0581.880.0581.860.0471.5040.0912.430.0721.920.0601.6150.0721.710.0621.460.0511.2060.0771.740.0671.520.0561.2770.1202.600.1252.700.1032.22Horizontal excitation (horizontal)30.0792.570.0702.290.0611.9940.0942.600.0932.540.0802.2050.1012.510.1052.600.0872.1560.1343.200.1032.450.0892.1170.1563.810.1172.860.0942.30Inclined excitation (vertical)30.1724.390.1132.890.0962.4540.1983.610.1282.340.1051.9250.2203.630.1362.230.1161.9160.2583.950.1592.420.1352.0670.2754.180.1682.550.1362.06Inclined excitation (horizontal)30.1165.320.0562.550.0462.1040.0922.990.0431.400.0361.1850.1003.080.0611.890.0501.5560.1193.720.0782.420.0652.0270.1705.210.1474.540.1223.76Average0.1353.260.0982.370.0821.98


## 5.3.2. Tension Forces

The displacements obtained at location 3 were used to estimate the tension force of cable 1, and those from location 5 were used to estimate the tension forces at cables 2 and 3. Figure25 presents the tension force of cable 3 under inclined excitation. A good agreement between the estimated and reference tension forces was observed with 11.20 NRMSE and 3% NRMSE. The tension forces estimated for all the cases are summarized in Figure 26. The proposed technique successfully estimated the tension forces, and RMSE and NRMSE values were less than 12 N and 4.6%, respectively. It should be noted that the load cells installed on cables 1 and 2 leaked after horizontal excitation, resulting in the failure of the reference tension force measurement under inclined excitation. Therefore, the corresponding NRMSE values were not calculated.Figure 25
Comparison of the tension force of cable 3 estimated by the proposed technique and measured by a load cell under inclined excitation in the 8 m-long SFT mock-up structure test.Figure 26
Cable tension force estimation error obtained from the 8 m-long SFT mock-up structure test.
(a)(b)


## 6. Conclusions

This study proposes a simultaneous displacement and cable force estimation technique for SFTs using acceleration and strain measurements. The main contribution of this study lies in the derivation of the strain-displacement transformation relationship for SFTs with simplified mode shapes, automatic mode-scaling factor estimation, and displacement-based mooring cable tension force estimation. The feasibility of the proposed technique was first examined on a numerical SFT model by estimating the bidirectional displacements at thirty-one locations and tension forces at 30 mooring cables. The NRMSE values of the displacements and tension forces estimated using the proposed technique were less than 1% and 2%, respectively. Experimental validation was conducted on an 8 m-long aluminum SFT mock-up structure. The overall RMSE and NRMSE values of the estimated displacements were 0.082 mm and 1.98%, respectively. In addition, tension forces of three mooring cables were estimated with RMSE and NRMSE values less than 12 N and 4.6%, respectively. The validation tests performed showed that the proposed technique can compensate for the discrepancy between the true and simplified mode shapes at most locations, and then accurately estimate the SFT displacements. However, it may not work well near the two ends of the tunnel, particularly when the boundary conditions are not symmetric. In addition, in the laboratory test, the SFT mock-up structure was excited by manually pushing and releasing the tunnel due to its relatively large stiffness. Further studies may be needed to examine the proposed technique under more realistic loading, such as vehicle and wave loading. In such a case, a longer SFT structure with less stiffness than the one investigated in this study is required to generate meaningful vibration.



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*Source: 7803876-2023-02-09.xml*