IFEM method for distributed monitoring of cable forces in cable-stayed bridges

Abstract

The method proposed in this article pertains to developing a technique for structural health monitoring of cable-stayed bridges based on the inverse finite element method (iFEM). This approach is built on distributed monitoring of strains by using a Brillouin scattering-based distributed optical fiber sensor. Using this method, both the deflections and changes in cable forces are computed to assess the structural health of bridges. In contrast to the existing distributed strain-based techniques, the computation of cable forces does not require prior knowledge of loads and their locations. This provides the opportunity for use under operational conditions of the bridge, where the bridge is subjected to moving vehicular loads. The capability of the proposed technique was evaluated in laboratory experiments by a scaled model of a cable-stayed bridge. The experiments involved static and dynamic loads and the acquisition of distributed strains by a PPP-BOTDA optical fiber interrogation unit. Other sensor types, such as LVDT and FBG sensors, were employed in the experiments to validate the results of the proposed computational approach. In addition, the differences between the proposed approach and previous techniques are compared in terms of the computational approach, their attributes, and percentage errors in the computation of cable force variations.

Keywords

bridge operational loads cable forces cable-stayed bridge distributed fiber-optic sensors iFEM reference-free deflections SHM

Introduction

Among different types of bridges, cable-stayed bridges have been widely employed for mid-to-long-span designs. The cables are the primary load paths between the deck and the pylons, and their integrity is essential for the bridge’s structural health. Considerable attention has been devoted to cable force monitoring because damage in one or more cables alters the force distribution system that supports the bridge. This consideration was leveraged to develop anomaly detection strategies based on the variations in cable forces.^1–3 A survey of technical literature further reveals that monitoring cable forces allows for estimating fatigue and assessing the fatigue-related residual useful life of the structural elements. Some studies have demonstrated that the actual fatigue-related service life of the cables deviates significantly from the estimates provided by design codes, which are based solely on traffic forecasts.⁴ Hence, monitoring cable forces is essential for accurately estimating the residual service life of the cables.

Various direct measurement techniques have been developed for estimating cable forces in cable-stayed bridges. For example, Zhang et al.⁵ introduced a synchro-squeezing wave-packet method, based on an instantaneous frequency tracking algorithm, to determine the cable forces using acceleration measurements of the main cables. Results from their research indicated that their approach can effectively identify time-varying cable tension changes during typhoons, and therefore, is applicable for real-time monitoring of cable tension under extreme operational conditions. On the other hand, Chen et al.⁶ investigated the suitability of specific features of the wind, such as speed, gust, and direction, from which it was possible to compute the cable forces from the accelerometer data reliably.

Other methods that involve the use of discrete sensors pertain to strain gauge-based techniques, such as the work reported by Nazarian et al.,⁷ who used strain gauges to measure and correlate the change in support reactions to loss of tension in the cables of the bridges. However, from a practical point of view, large numbers of externally attached strain gauges and accelerometers are required to monitor all cables within cable-stayed bridges effectively. Using many discrete sensors and running lead lines along the lengths of the bridges is costly in terms of time, labor, and materials. Large volumes of sensor data must be synchronized and processed, considering the number of sensors attached to the cables. Furthermore, sensors externally attached to the cables primarily apply only to the cables that do not use corrosion protection casings. Considering all the abovementioned limitations, using a single distributed sensor is more practical and cost-effective.

Vision-based approaches have partially overcome this limitation, but their applicability is limited to favorable weather conditions, as fog or insufficient lighting may severely affect the image quality.⁸ A few examples are provided herein for completeness. For instance, Zhao et al.⁹ developed microwave interferometric radar to capture the vibration of the cables, then the cable forces were converted from the vibration data. Jo et al.¹⁰ proposed a method for evaluating the level of cable tension in cable-stayed bridges using multiple digital images. Du et al.¹¹ explored digital image processing techniques in cable force monitoring. In their study, the cable vibration was obtained from images using digital image processing and digital image correlation, and then the vibration frequencies were correlated to cable forces.

Cable-stayed bridges are generally flexible structures,¹² where the bridge deck offers valuable insights into the condition of the stay cables. Several studies were initiated to alleviate sensor installation accessibility issues by indirectly monitoring the cable forces from the deck response. Liu et al.¹³ presented a simplified equation to relate changes in cable forces to girder displacements. A bridge was employed as a case study and confirmed the method’s accuracy with the use of 22 displacement sensors. Saeed et al.¹⁴ presented a straightforward method to control nodal displacements and internal forces in cable-stayed bridge decks by calculating the necessary length adjustments for the cables. The technique is theoretically and experimentally supported using a laboratory-built single-tower bridge model with a displacement gauge under each cable anchor location. Wang et al.¹⁵ proposed a new method to localize and quantify partial cable damage in cable-stayed bridges by analyzing abnormal temperature-induced change in girder deflections. A mechanical model was employed to link the deflections to cable forces. A cable-stayed bridge was employed to validate the method by using eighteen displacement sensors along the length of the bridge.

The studies mentioned above demonstrate that the amplitudes of forces exerted by cables have a measurable impact on deck displacements. As the cable force is estimated indirectly through deck deflections, the accuracy of deflection-based methods relies on the resolution of deflection measurement methods. Directly measuring deflection in the field can be difficult, particularly on bridges lacking fixed reference points. To overcome this, researchers have investigated reference-free displacement estimation methods. Helmi et al.¹⁶ used tiltmeters and strain gauges for bridge deflection monitoring by assuming the deflection profile is a polynomial function. Ma et al.¹⁷ proposed a method to estimate bridge displacements by fusing strain and acceleration data using a filter and a recursive algorithm. Park et al.¹⁸ proposed an indirect method using both acceleration and strain data to estimate nonzero mean, dynamic displacements. Shin et al.¹⁹ introduced an algorithm that estimates displacements using vibration strains from FBG sensors and theoretical mode shapes of a simply supported beam. Skafte et al.²⁰ introduced a strain-based method for estimating dynamic displacements using operational modal analysis techniques. Despite avoiding fixed reference points, these approaches still require dense sensor deployment, raising wiring complexity, cost, and large-scale data processing issues, especially in dynamic monitoring scenarios.

Distributed fiber-optic sensors have become a prevalent technology for addressing these issues.^21–24 Brillouin scattering-based fiber-optic systems provide continuous strain measurements along the entire length of the optical fiber with a measurement range extending to 30 km or more.^25,26 Depending on the optical interrogation technology, these sensors offer spatial resolutions ranging from 5 cm to 1 m.^27–35 A limited number of investigators have studied the applications of distributed fiber-optic sensors for cable forces monitoring through deck response. Nazarian et al.³⁶ introduced an approach using a Brillouin optical time domain analyzer (BOTDA) as a distributed sensor to detect tension loss in the cables of cable-stayed bridges. The method was based on the interrelationship between individual cable forces and the support reactions. Scarella et al.³⁷ extended the formulations of Nazarian et al.’s approach for monitoring the cable forces for bridges subjected to dynamic forces. Both methods were based on the flexibility approach for the analysis of structures. The analysis approach involves discretizing the external loads carried by the cables into a set of concentrated forces on the deck. Then, a linear system of equations was established to relate the measured strains to the discretized external loads. This technique requires the number of measurement points to be larger than the number of reconstructed loads, and the sensor locations have to be well designed to ensure the linear system of equations is well conditioned.^38,39 However, the flexibility approach is not applicable when the locations and the number of external loads applied to the structure are unknown,⁴⁰ that is, when the bridge is under normal traffic operations.

Despite the practical limitations of the method based on the distributed monitoring of bridge deck strains, the studies mentioned above showed the potential for indirectly monitoring cable forces from the distributed bridge deck strains. Instead of the flexibility approach, deck deflections could be an intermediate variable linking the distributed strains along the bridge deck and cable force, eliminating the need for external load information. Various computational approaches exist for determining structural deflections from distributed strain data, such as the modal shape method, the direct integration of strains, and the inverse finite element method (iFEM). The modal shape method uses mathematical functions of fundamental vibration modes to fit displacement or strain fields. The coefficient that represents the displacement is identified by minimizing the discrepancy between the calculated and measured strains. Accurate material property data and a validated model (usually a finite element model) are needed for modal analysis.^41–45 Alternatively, the direct strain integration technique reconstructs displacements by double integrating measured strain values, assuming beam-like behavior. However, this approach tends to be susceptible to noise, leading to cumulative errors over extended lengths.^46,47 A detailed evaluation by Gherlone et al. highlighted the iFEM as a particularly effective and versatile method.⁴⁸ Its strengths lie in its adaptability to different structural geometries and support conditions and its independence from external loads and material properties. Song et al.⁴⁹ proposed a real-time method for predicting dynamic deformation of beam structures using FBG strain sensors, combining Kalman filtering and inverse finite element method (iFEM). Since iFEM computes the displacements using a least squares algorithm, the distributed strain measurement points provided by BOTDA will reduce error and improve robustness against noise. These attributes make the combination of BOTDA and iFEM particularly suited for deflection estimation and further determination of cable forces.

The research described in this article aimed to establish a method to estimate cable forces in bridges under arbitrary loads using distributed optical fiber sensors. The novelty of the proposed approach lies in combining iFEM with an analytical structural model to monitor all cable forces without requiring knowledge of external force locations—a key limitation of previous studies, which limits their applicability in real-world scenarios where bridge traffic is inherently unpredictable. The proposed method was validated using a laboratory-scaled model of the Dongshuimen Bridge in Chongqing, China. Dongshuimen Bridge is a three-span harp-patterned cable-stayed bridge. Both static and dynamic tests were conducted in this research to evaluate the performance of the proposed method. The methodology implementation involved using a pulse-pre-pump Brillouin optical time domain analysis system (PPP-BOTDA), referred to as BOTDA, to monitor the strain distribution along the bridge spans. The inverse finite element method (iFEM) was applied to convert the distributed strains into distributed deflections. Then, the cable force variations were computed using an analytical formulation of the interrelationship between deck deflection and cable forces. The formulation of the method developed herein is described next, followed by the description of the experimental program and the analysis of the results.

Methodology

The process for implementing the method introduced herein involves the computation of the distributed deflection along the length of the bridge and relating the individual cable force changes with the displacement of the deck at the locations of the cables. The iFEM is employed first to compute deflections from distributed strains in the formulation of this method. Then, the cable forces are evaluated based on the deflections at the locations of the cables on the deck by taking advantage of the compatibility of cable-deck displacements.

A brief introduction to iFEM

The iFEM, originally developed by Tessler and Spangler,⁵⁰ is a variational approach for estimating full-field deflections from strain measurements without requiring knowledge of the applied load. The method involves discretizing the structure into inverse finite elements and minimizing a functional defined as the error between the measured strain and those derived from an unknown deflection field. This field is defined by the element shape functions and the nodal degrees of freedom. The minimization procedure is analytical and leads to a linear system of equations with nodal deflections as unknowns. Once the boundary conditions are applied, the linear system of equations is solved, then the full-field deflection is reconstructed via the element shape functions.

In this study, the iFEM formulation described is related to the Euler–Bernoulli inverse beam element, illustrated in Figure 1. A global Cartesian reference system ( $X, Y, Z$ ) is defined as the main reference system for all the elements. The element is denoted with the superscript $\cdot^{e}$ and is considered isotropic with known area, $A^{e}$ , moment of inertia with respect to the principal $x$ - and $y$ -axes, $I_{x}^{e}, I_{y}^{e}$ , and length $L_{e}$ . The element is based on two nodes (node 1 and node 2 in Figure 1) and it is described using a local Cartesian coordinate system $(x, y, z)$ as shown in Figure 1, where $u, v, w$ are the longitudinal deflections along the axes and $ϕ$ refers to rotation around the axes. A global Cartesian reference system $(X, Y, Z)$ is also defined in Figure 1: it is the reference system for the whole model. The subscripts $\cdot_{1}$ and $\cdot_{2}$ in Figure 1 denote the degrees of freedom of nodes 1 and 2, respectively.

Applying the small angle approximation, the displacement field of the beam is expressed as follows:

\begin{matrix} u_{z} (x, y, z) = w (z) + y ϕ_{x} (z) - x ϕ_{y} (z) \\ u_{y} (x, y, z) = v (z) \\ u_{x} (x, y, z) = u (z) \end{matrix}

(1)

Where $u_{x}$ , $u_{y}$ , $u_{z}$ refer to the displacements field along the $x$ -, $y$ -, and $z$ - axes. For convenience, the kinematic variables of the displacements field are collected into the vector $u = [u, w, v, ϕ_{x}, ϕ_{y}]$ .

The strain field can be derived from equation (1) according to the small-strain hypothesis theory, as shown

\begin{matrix} ϵ_{z} (x, y, z) = e_{1} (z) + y e_{2} (z) - x e_{3} (z) \\ γ_{yz} (x, z) = e_{4} (z) \\ γ_{xz} (y, z) = e_{5} (z) \end{matrix}

(2)

where $ϵ_{z}$ is the strain in the $z$ -direction, $γ_{yz}$ and $γ_{xz}$ are the shear strains in the $yz$ and $xz$ planes, respectively. The section strains vector $e (u) = [e_{1}, e_{2}, e_{3}, e_{4}, e_{5}]$ is defined as follows:

\begin{matrix} e_{1} = \frac{dw}{dz} (z) \\ e_{2} = \frac{d ϕ_{x}}{dz} (z) \\ e_{3} = \frac{d ϕ_{y}}{dz} (z) \\ e_{4} = \frac{dv}{dz} (z) + ϕ_{x} (z) \\ e_{5} = \frac{du}{dz} (z) - ϕ_{y} (z) \end{matrix}

(3)

The section strain vector is defined as in Gherlone et al.,⁵¹ and contains curvatures, shear strains, and axial strain of the beam.

The inverse finite element method (iFEM) reconstructs the deformed shape of a structure by minimizing the following least-squares functional, which quantifies the error between the experimentally measured section strains $e^{m}$ and the iFEM model section strains $e (u)$ . From this point forward, the superscript $\cdot^{m}$ will denote experimentally measured quantities:

Φ = ‖ e (u) - e^{m} ‖^{2}

(4)

The kinematic variables of the displacement field in equation (1) are discretized with finite elements based on $C^{0}$ -continous shape functions,⁵¹ denoted with the matrix $N_{e} (z)$ :

u_{e} (z) = N_{e} (z) u_{e}

(5)

Where $u_{e}$ contains the nodal degrees of freedom of the element, and $z$ refers to the z-axis of the local Cartesian coordinate system. From this point forward, the subscript $\cdot_{e}$ will denote variables related to the e-th element. Following Equations (2) and (3), the section strains $e (u)$ . are linearly related to the nodal degrees of freedom of the element $u_{e} = {[w_{1}, v_{1}, ϕ_{x_{1}}, u_{1}, ϕ_{y_{1}}, w_{2}, v_{2}, ϕ_{x_{2}}, u_{2}, ϕ_{y_{2}}]}^{T}$ via the element shape function derivatives $B_{e} (z)$ :

e_{e} (u_{e}) = B_{e} (z) u_{e}

(6)

Consequently, the least square functional in Equation (4) is the summation of the contributions of all the $N$ elements:

Φ = \sum_{e = 1}^{N} Φ_{e} = \sum_{e = 1}^{N} ‖ e_{e} (u_{e}) - e_{e}^{m} ‖^{2}

(7)

Minimizing Equation (7) leads to a linear system of equations, where the unknowns are the nodal displacements of the elements. After incorporating problem-specific boundary conditions, this system of equations is solved to obtain nodal displacements. Using the element shape functions, the displacement at any point within the structure can then be reconstructed. The next paragraph illustrates this procedure in detail, presenting a simplified version of the general iFEM formulation tailored to the application analyzed in this article.

Cable force estimating methodology

Deck deflection calculation through distributed strain

The method for deck deflection calculation presented in this article builds on the authors’ previous research.⁵² In this work, it has been adapted for cable force monitoring by extracting deck deflection at specific locations. Computing the distributed deflections from the distributed strains involves instrumenting the bridge deck with a distributed fiber-optic sensor, as schematically shown in Figure 2(a), and then using the iFEM to convert the distributed strains to distributed deflections. In modeling the bridge in an iFEM model, the bridge deck is discretized with $N$ inverse beam elements, as shown in Figure 2(b). Figure 2 pertains to the general description of the discretization process, as the iFEM method works on any boundary condition. The Euler–Bernoulli inverse beam element is employed for the construction of the iFEM model.⁵³ Each inverse beam element is as depicted in Figure 1. The degrees of freedom for each element are represented by the vector $u_{e} = {[w_{1}, v_{1}, ϕ_{x_{1}}, u_{1}, ϕ_{y_{1}}, w_{2}, v_{2}, ϕ_{x_{2}}, u_{2}, ϕ_{y_{2}}]}^{T}$ , which contains the nodal displacements (or deflections).

Considering the bridge as the modeling object, according to the Euler–Bernoulli kinematic theory, the only nonnull strain component is a strain in the z direction. Therefore, only $ϵ_{z} (x, y, z)$ remains in Equation (2). Furthermore, because the primary focus is on the vertical deflection of the bridge deck in flexure, the axial and lateral strain terms do not contribute to the vertical deflections, and they are omitted from this formulation. According to these assumptions, the section strains $e_{e} (u_{e})$ and $e_{e}^{m}$ can be simplified to $e_{e} (u_{e}) = [e_{2} (u_{e})]$ and $e_{e}^{m} = [e_{2}^{m}]$ , respectively, given that $e_{2}$ represents the vertical curvature of the neutral axis in Equation (3). Accordingly, $Φ_{e}$ in Equation (7) can be simplified:

\begin{matrix} Φ_{e} (u_{e}) = \frac{I_{x}^{e} L^{e}}{A^{e} n_{e}} \sum_{i = 1}^{n_{e}} {(e_{2} (z_{i}) - e_{2_{i}}^{m})}^{2} \\ = \frac{I_{x}^{e} L^{e}}{A^{e} n_{e}} \sum_{i = 1}^{n_{e}} ({(e_{2} (z_{i}))}^{2} - 2 e_{2} (z_{i}) e_{2_{i}}^{m} + {(e_{2_{i}}^{m})}^{2}) \end{matrix}

(8)

where $I_{x}^{e}$ is the element’s second area moment, $L^{e}$ is the element length, $A^{e}$ is the element area, and, $n_{e}$ is the number of measurements inside the e-th element. The curvature measurements within the elements are $e_{2_{i}}^{m}$ , while the iFEM curvatures $e_{2} (z_{i})$ are functions of the nodal displacements $u_{e}$ . The curvature measurements are directly obtained from strain measurements taken on the bottom or the top extreme fibers of the bridge girder (see Figure 2 (a)), that is, $e_{2_{i}}^{m} = \frac{ϵ_{i}}{d}$ , where $ϵ_{i}$ is the measured longitudinal strain on the bridge girder and, $d$ , the distance between the measurement point and the section neutral axis.

The iFEM model strain (which are still unknown at this derivation step) can be evaluated at the locations where each experimental measurement $e_{2_{i}}^{m}$ is available by utilizing the element shape functions derivatives,⁵³ that is, substituting ( $e_{2} (z_{i}) = B_{χ_{x}} (z_{i}) u_{e}$ ) in Eq. (8). Expanding each square element, the $Φ_{e}$ in Equation (8) can be rearranged as follows:

\begin{matrix} Φ_{e} = u_{e}^{T} (\frac{I_{x}^{e} L^{e}}{A^{e} n_{e}} \sum_{i = 1}^{n_{e}} B_{χ_{x}} {(z_{i})}^{T} B_{χ_{x}} (z_{i})) u_{e} \\ - u_{e}^{T} (2 \frac{I_{x}^{e} L^{e}}{A^{e} n_{e}} \sum_{i = 1}^{n_{e}} B_{χ_{x}} {(z_{i})}^{T} \frac{ϵ_{i}}{d}) + c \end{matrix}

(9)

where $B_{χ_{x}} {(z_{i})}^{T}$ is the element shape function derivative evaluated at the coordinate of the $i$ -th measurement point inside the element, and c is a constant invariant to the displacements. Equation (9) can be simplified as follows:

Φ_{e} = u_{e}^{T} k_{e} u_{e} - u_{e}^{T} f_{e} + c

(10)

where $f_{e} = 2 \frac{I_{x}^{e} L^{e}}{A^{e} n_{e}} \sum_{i = 1}^{n_{e}} {B_{χ_{x}} (z_{i})}^{T} \frac{ϵ_{i}}{d}$ is a column vector that depends on the measured BOTDA strain data $ϵ_{i}$ of the element, $k_{e} = \frac{I_{x}^{e} L^{e}}{A^{e} n_{e}} \sum_{i = 1}^{n_{e}} {B_{χ_{x}} (z_{i})}^{T} B_{χ_{x}} (z_{i})$ is the compatibility matrix, which only depends on the element’s geometry. Once each single contribution for each element, $Φ_{e}$ , has been written, it can be rotated in a global reference system. These contributions can be assembled by the standard assembly procedure for finite elements by inserting Equation (10) into Equation (7), yielding the following linear functional:

Φ = U^{T} K U - U^{T} F + c

(11)

where the vector $U = [u_{1}, u_{2}, . ., u_{N}]^{T}$ is the collection of the degrees of $u_{e}$ , in the global reference system, the vector $F = [f_{1}, f_{2}, . ., f_{N}]^{T}$ is the collection of $f_{e}$ in the global reference system, and $K$ is assembled from $k_{e}$ for all the elements. This functional, $Φ$ , is minimized with respect to $U$ , and problem-specific boundary conditions can be applied. This results in a linear system of equations:

K_{B} U_{B} = F_{B}

(12)

where the subscript $\cdot_{B}$ , indicates that the boundary conditions have been applied, the matrix $K_{B}$ depends only on the element geometry, and it is not singular, provided that at least two curvature measurements are given for each element⁵³ and that the applied boundary conditions statically constrain the structure, that is, it is not free to move. The term $F_{B}$ depends only on the measured curvatures and the element’s geometry. Hence, the iFEM requires no preknowledge of the external loads or the structural dynamic properties, such as inertia, damping, and stiffness. By solving the linear system of equations in Equation (12), the nodal deflections of the deck $U_{B}$ can be computed. From the nodal deflections, computed distributed deflections can be computed through the element shape functions. The distributed deck deflections are then employed to compute the cable force changes, which is described next.

Cable force calculation through deck deflection

The change in cable forces is correlated with the deflection-induced elongations of the cables. The deflection of the deck at the locations of the cable anchors is directly related to the cable elongation. Therefore, only the deck deflections at the locations of the cables $V_{i}$ were extracted from distributed deflections $U_{B}$ , as shown in Figure 3. Figure 3 corresponds to a Pylon with several cables designated by $C$ . Neglecting the cable and pylon inertias, and considering no external loads are applied to the piers, the relationship between the deck vertical displacement and the force change in a cable is described by the following equation, initially developed by Hegab⁵⁴:

T_{i} = {\begin{matrix} k_{i} (- V_{i} \sin (θ_{i}) + H_{i} \cos (θ_{i})), i \leq C / 2 \\ k_{i} (- V_{i} \sin (θ_{i}) - H_{i} \cos (θ_{i})), i > C / 2 \end{matrix}

(13)

where $T_{i}$ is the force change in cable $i$ , $k_{i} = \frac{E_{i} A_{i}}{L_{i}}$ is the cable stiffness ( $E_{i}$ is the elastic modulus of cable $i$ , $A_{i}$ its cross section and $L_{i}$ its length), $V_{i}$ is the deck vertical deflection at the location of cable $i$ , $H_{i}$ is the Pylon’s horizontal displacement at the location of the cable $i$ , and $θ_{i}$ is the angle between the deck and cable $i$ . These parameters are shown in Figure 3. Cable sag is also neglected in Equation (13) because, during monitoring, the focus is on relative changes in tension rather than absolute values, and the contribution from the self-weight (which induces the sag) of the cable assumed to remain constant after construction.

The only unknown in Equation (13) is the Pylon horizontal deflection $H_{i}$ , which is a function of the force changes in all the attached cables. $H_{i}$ can be computed as follows³⁶:

H_{i} = \sum_{j = 1}^{C / 2} (T_{j} \cos (θ_{j}) - T_{C + 1 - j} \cos (θ_{C + 1 - j})) g_{ij}

(14)

where $C$ is the total number of cables connected to the Pylon. The term $g_{ij}$ is a flexibility coefficient representing the Pylon deflection at the location of cable $i$ due to a unit force applied at the location of cable $j$ . By modeling the Pylon as a cantilever beam $g_{ij}$ can be computed based on the following relationship⁵⁵:

g_{ij} = {\begin{matrix} \frac{h_{j}^{3}}{3 EI}, h_{i} = h_{j} \\ h_{j}^{2} \frac{3 h_{i} - h_{j}}{6 EI}, h_{i} < h_{j} \\ h_{i}^{2} \frac{3 h_{j} - h_{i}}{6 EI}, h_{i} > h_{j} \end{matrix}

(15)

where $h_{i}$ and $h_{j}$ are the distances between the Pylon and the location of cables $i$ and $j$ on the Pylon, as shown in Figure 3 (a).

With $H_{i}$ and $g_{ij}$ computed by Equations (14) and (15), the cable force changes $T_{i}$ in Eq. (13) can be expressed as a function of the forces in all the cables and by the vertical deck deflections $V_{i}$ at the locations of the cables on the deck. The change in the cable forces for all cables of the bridge can be expressed by Equation (13) and reorganized to build a linear system of equations as shown below:

An = v

(16)

where the vector $n = [T_{1}, \dots, T_{C}]^{T}$ contains the cable force changes, the vector $v = [v_{1}, \dots, v_{C}]^{T}$ comprises the scaled deck displacements at the locations of the cables on the deck, and $A$ is a square flexibility matrix containing the stiffness and geometric parameters of the wires and the Pylon. The elements in the vector $v$ can be expressed as follows:

v_{i} = - k_{i} V_{i} \sin (θ_{i})

(17)

The elements of the flexibility matrix $A$ , $A_{ij}$ , represent the vertical deflection of the deck at cable $i$ due to a unit force change in cable $j$ . It can be expressed as follows:

A_{ij} = {\begin{matrix} k_{i} g_{i, j} \cos {(θ_{i})}^{2} + 1 if i = j \\ k_{i} g_{i, j} \cos (θ_{i}) \cos (θ_{j}) if i \neq j, i \leq \frac{C}{2}, j \leq \frac{C}{2} \\ - k_{i} g_{i, C - j + 1} \cos (θ_{i}) \cos (θ_{j}) if i \neq j, i \leq \frac{C}{2}, j > \frac{C}{2} \\ - k_{i} g_{i, j} \cos (θ_{i}) \cos (θ_{j}) if i \neq j, i > \frac{C}{2}, j \leq \frac{C}{2} \\ k_{i} g_{i, C - j + 1} \cos (θ_{i}) \cos (θ_{j}) if i \neq j, i > \frac{C}{2}, j > \frac{C}{2} \end{matrix}

(18)

The cable force changes are computed by solving Equation (16).

Experimental program

The capability of the proposed method for monitoring the change in cable forces was evaluated by testing a scaled model of the Dongshuimen Bridge, a railroad cable-stayed bridge crossing the Yangtze River in Chongqing, China. The Dongshuimen Bridge is a double-tower cable-stayed steel truss bridge with three spans of 222.5 m, 445 m, and 190.5 m, respectively. The model of the bridge shown in Figure 4 was scaled using the elastic direct method with a scaling factor of 1/60.⁵⁶ A 50.8 × 50.8 mm hollow-box shape steel was selected to replicate the boxed truss deck of the prototype, with a wall thickness of 3.81 mm. The two Pylons were fixed at the base with a moment of inertia of 132,248 mm⁴. The deck and the Pylons were made of steel with a yield strength of 235 MPa. The cables were made of steel piano wires with a diameter of 0.4 mm and a tensile strength of 2500 MPa. Turnbuckles were employed to control the tension in the individual cables of the bridge, as shown in Figure 5. Lumped masses were added at equal intervals to the deck to satisfy the dead load scaling requirements. In addition, the cable forces were pretensioned to 145–164.6 kN according to the design document and the scaling factor.

A single-mode telecommunication grade optical fiber (SMF-28) manufactured by Corning⁵⁷ was used for distributed strain sensing by a BOTDA. The optical fiber was adhered to the top surface of the bridge deck. Fiber-optic Bragg grating sensors (FBGs) were adhered to the eastward cables of the west Pylon for cable force measurements. In addition, three LVDTs were employed to measure the deflection of the bridge deck. The process for adhering the optical fiber for distributed measurement of strains to the deck involved cleaning and sanding the steel surface of the deck. The optical fiber was adhered to the top face of the bridge deck using a silicon-based adhesive, as shown in Figure 5. According to Equation (13), the cable force is controlled by the deflection at the locations where the cables are anchored to the deck. For this reason, the optical fiber was adhered as close to the cable anchor line along the bridge’s surface. The distributed strain was measured using a BOTDA interrogator manufactured by Neubrex (NBX-6055), with a measurement accuracy of 10 με. The effectiveness of the Brillouin-scattering-based sensing system for distributed strain measurements depends on the system’s spatial resolution (SR) and the sampling interval (SI). SR refers to the length of the segment where the system calculates the average strain along the fiber, and SI is the distance between successive measurement points along the fiber. A SR of 10 cm with an SI of 1 cm was employed in this research. Three LVDTs were installed at the midspans of each bridge span to cross-verify deflections, as shown in Figures 4 and 5(b).

In addition to the distributed sensor, as shown in Figure 5(c), nine FBG sensors were adhered to the cables using epoxy to directly measure cable forces and validate the cable force computations derived from the distributed strains. Compared to the Brillouin scattering-based sensing systems, such as the BOTDA employed herein, FBGs provide higher resolution measurements but can only make discrete measurements. More details on FBG sensors can be found in this article by Xu et al.⁵⁸ The FBG instrumented cables are shown in Figure 4 and pertain to cable numbers 10 to 18. A micron optics interrogator unit was used for data acquisition by transducing the FBG sensors’ signals. The FBGs were calibrated to measure cable forces utilizing a dynamometer.

The iFEM model of the bridge girder, shown in Figure 6, comprises 52 inverse beam elements. The girder was pinned at the tower locations and the two abutments. The pylons were modeled using beam elements, and the cables were modeled as components only subjected to axial deformations. The elastic modulus of the steel elements employed for both the cables and Pylons was 210 GPa. Based on the material and geometrical properties of the pylons and the cables listed above, all the required parameters to compute matrix $A$ in Equation 16 were determined. The resulting matrix 18 × 18 full rank square matrix enabled the computation of cable tension change from the vertical deck deflections estimated by the iFEM using the distributed strain measurements.

The experimental program was designed to evaluate the effectiveness of the proposed method for static and dynamic loads. The static tests involved placing calibrated loads at combinations of two different loading positions, thus showing the proposed method’s capability of reconstructing the changes in the cable forces independent of the magnitudes and their locations. The dynamic test included the excitation of different vibration modes of the bridge to evaluate its effectiveness under operational conditions.

Static tests

In the static experiments, ten-kilogram weights were applied via strings at specific locations on the specimen to simulate loadings representative of the bridge’s gross vehicle weight (GVW). According to the scaling theory for elastic modeling described by Harris and Sabnis,⁵⁶ 10 kg weight in the scale model corresponds to 352.8 kN on the actual bridge, given the adopted scaling factor of 1/60. This is aligned with the average truck gross weight of 360 kN as reported by Fu et al.⁵⁹ The specific load locations and combinations are illustrated in Table 1 and in Figure 7. Figure 8 shows a typical load applied to the structure at the load location designated B.

Table 1.

Test names with load locations and total weight attached to the bridge deck.

Test Name	Location A	Location B
Static-1		10 kg
Static-2	10 kg
Static-3	10 kg	10 kg

A sampling interval of 1 cm at a spatial resolution of 5 cm was selected to monitor the bridge deck’s distributed strain with a PPP-BOTDA interrogator.⁶⁰ Figure 9 pertains to the monitored distributed strain measurements along the cable-stayed bridge deck collected during static test number 2.

The measured distributed strains did not require any data smoothing process, and they were employed in the iFEM model, because the iFEM shape functions and mesh density automatically serve as a smoothing function, reducing the noise. Knowing the distance between the sensor location and the neutral axis (taken as half the height of the cross section in the experiment), the deflection along the bridge was calculated by solving Equation (12). For field applications, the location of the neutral axis can be determined either from design plan documents or experimentally computed by placing BOTDA sensors at two different levels on the bridge deck (e.g., on both the top and bottom surfaces of the deck). The LVDT measurements were employed to evaluate the distributed deflections. Figure 10 corresponds to the comparison between the distributed and LVDT measurements of the deflections. Figure 10 shows the differences between the distributed and the LVDT measurements for the three load cases. As summarized in Table 2, the differences between the calculated and measured deflections were 9.1%, 11.9%, and 6.7% for tests 1, 2, and 3, respectively. The distributed displacements estimated from iFEM were then used to calculate the cable force changes due to the application of the weights. The displacement at each cable position was acquired from the distributed displacements and employed in Equation (13). The cable forces were computed by using Equation (16). The changes in cable force measured by the FBG and calculated with the proposed method are compared in Figure 11. Table 2 specifies the values of the relative error between the measured and computed cable forces.

Table 2.

Difference between computed and measured results.

Test name	Root mean square and relative difference in deflection	Root mean square and relative difference in cable force
Static-1	0.19 mm (9.1%)	0.73 N (12.3%)
Static-2	0.14 mm (11.9%)	0.22 N (12.4%)
Static-3	0.11 mm (6.7%)	0.65 N (7.1%)

The deck deflections showed an average relative difference of 9.2%, whereas the cable forces demonstrated a slightly higher average relative difference of 10.6%. The difference between the two calculations was attributable to the error accumulation from estimating the vertical deflection into the cable tensions, the noise of the BOTDA measurement system, and inevitable modeling inaccuracies. The cause of the larger errors observed in cable 11 remains unclear. Possible reasons could be measurement noise or a sensor installation issue. For cables 17 and 18, the discrepancies are attributed to second-order effects becoming significant, particularly the influence of deck section rotation in addition to the vertical deflection. These effects are more pronounced in the outer cables, where section rotations are larger and the cables are more horizontally oriented, thus increasing their contribution to the measured force. Other potential sources of error may correspond to the sliding of the turnbuckles on the eyebolts anchoring the cables to the deck.

Dynamic test

During the dynamic test, the BOTDA strains were acquired in the amplitude transfer (AT) mode of the PPP-BOTDA system.⁶¹ The experiment was performed by analyzing the cable force variation under the dynamic motion of the scaled bridge deck. The experimental program involved oscillating the bridge deck vertically. The 10 kg weight representative of the real bridge GVW, involved in the static tests was also applied at location B (see Figure 7). The application of the representative GVW aimed to reproduce real scenarios where variations in cable forces can result from vibrations induced by operating traffic conditions on the bridge. After hanging the weight to the deck with a stiff cord, the vibrations were accomplished by manually shaking the mid-section of the bridge for 20 s, after which the bridge vibrated freely for 2 s. Then, five more vibration events were induced by pushing down and releasing the bridge span. These experiments were performed to apply different types of excitations to the bridge (i.e., forced and free vibrations). Although the dynamic test was not entirely representative of a real-world operational scenario of the bridge, it was instrumental in estimating the performance of the proposed method under different types of operational scenarios.

Figure 12 shows the measured strains for the dynamic tests. The measured strains were acquired at a sampling rate of 30 Hz, with a spatial resolution of 10 cm and a sampling interval of 5 cm. LVDTs and FBGs were acquired at 100 Hz. All the measured signals were resampled at 10 Hz using antialiasing filters to synchronize BOTDA and LVDT measurements and for denoising purposes. This process facilitated a comparison between the computed and LVDT-measured deflections.

The results of the dynamic test are presented in Figures 13 and 14. The deflections estimated by the iFEM model are compared with the LVDT measurements in Figure 13. Figure 13 pertains to the comparison of the time domain computed deflections at the location of LVDT-1. Similarly, in Figure 14, the computed and FBG-measured cable forces for cable numbers 13 and 15 are compared in the time domain.

The performance of the proposed method during the dynamic test was then quantified using the coefficient of correlation as summarized in Figure 15. The deflection estimation at the LVDT locations presented a coefficient of correlation of 0.98. In contrast, the cable force changes presented a correlation coefficient of 0.89. The statistics of the difference between computed and measured results are provided in Table 3. The mean percentage differences of 15.4% and 17.5% were found in deflection and cable force change estimations, respectively.

Table 3.

Statistics of the difference between computed and measured results of the dynamic test.

Test name	Max Abs difference	Root mean square difference	Mean abs percentage difference
Cable Force	13.4N	1.45N	17.5
Deflection	2.96 mm	0.31 mm	15.4

Cable force measurements in static tests generally provided better results than dynamic test. This is due to higher measurement noise by the BOTDA in the dynamic mode. Some other likely sources of error pertain to the unintentional lateral displacements induced by manual shaking during the dynamic test. Despite the experimental differences, the results were generally consistent. Table 4 compares the method introduced here with previous studies. The numerical values presented in Table 4 are directly cited from previous works. The comparisons are made regarding measurement resolution due to cable force changes and error levels. The table embodies measurements in terms of static and dynamic experiments. The study by Nazarian etal.³⁶ was performed under static loading conditions, and the resolution of measurements was limited to cable force changes of 30% or more. In Scarella’s³⁷ study, error levels increased when cable force changes were below 50%, and Table 4 presents the error corresponding to a force change of less than 20%. The percentage errors in the present study pertain to the cable force changes under 6% for experiments where the bridge was subjected to static tests, and 20% for dynamic tests. The better performance of the method presented here is attributed to the role of the iFEM shape functions, and the mesh density having a profound effect in reducing the noise, as discussed earlier.

Table 4.

Comparison of the methods employed for distributed detection of cable force changes.

Study	Method	Requires prior knowledge of loads	Cable force (static)	Cable force (dynamic)	Monitors deflection	Error in static	Error in dynamic
Proposed method	iFEM+ analytical method	No	Yes	Yes	Yes	10.6%	17.5%
Nazarian, et al.³⁶	Flexibility method	Yes	Yes	No	No	50%	NA
Scarella, et al.³⁷	Flexibility method	Yes	No	Yes	No	NA	32.7%

Conclusion

The study reported in this article develops a novel method for monitoring the cable force changes in cable-stayed bridges using a single BOTDA fiber optic sensor under arbitrary loading conditions. The proposed method utilizes distributed strain measurements of the bridge deck to calculate distributed deck deflections using the iFEM method. In using this approach, the need for prior knowledge of loads and their locations is eliminated. The distributed deck deflections are subsequently utilized to compute cable force changes via an analytical model relating strains between the tower and the cable stays. The novelty of the method developed herein resides in its ability to monitor cable force changes due to arbitrary dynamic loads typical in bridges subjected to operational conditions using a single BOTDA fiber-optic sensor. It enables the computation of distributed deflections as well as cable force changes. The resolution of the measurements compared to the previous studies is much higher, both under static and dynamic loads. The performance of the proposed work for field applications needs to be further studied in future research. The focus is further developing the method by installing fiber-optic sensors along the sides of the bridge deck to measure lateral strains and compute lateral deflections. This would enable refinement of the cable force equations presented above by incorporating the effects of the bridge’s lateral movements.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Farhad Ansari

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