Damaged cable identification in cable-stayed bridge from bridge deck strain measurements using support vector machine

Abstract

A new two-step approach is developed for damaged cable identification in a cable-stayed bridge from deck bending strain responses using Support Vector Machine. A Damaged Cable Identification Machine (DCIM) based on support vector classification is constructed to determine the damaged cable and a Damage Severity Identification Machine (DSIM) based on support vector regression is built to estimate the damage severity. A field cable-stayed bridge with a long-term monitoring system is used to verify the proposed method. The three-dimensional Finite Element Model (FEM) of the cable-stayed bridge is established using ANSYS, and the model is validated using the field testing results, such as the mode shape, natural frequencies and its bending strain responses of the bridge under a moving vehicle. Then the validated FEM is used to simulate the bending strain responses of the longitude deck near the cable anchors when the vehicle is passing over the bridge. Different damage scenarios are simulated for each cable with various severities. Based on damage indexes vector, the training datasets and testing datasets are acquired, including single damaged cable scenarios and double damaged cable scenarios. Eventually, DCIM is trained using Support Vector Classification Machine and DSIM is trained using Support Vector Regression Machine. The testing datasets are input in DCIM and DSIM to check their accuracy and generalization capability. Different noise levels including 5%, 10%, and 20% are considered to study their anti-noise capability. The results show that DCIM and DSIM both have good generalization capability and anti-noise capability.

Keywords

damage identification bending strain support vector machine finite element model cable-stayed bridge

Introduction

Cable-stayed bridges are widely used all around the world. In this type of bridges, cables as the crucial components transfer the dead loads and live loads from the deck to the pylons. However, in practice, the cables are prone to deterioration and damage because of fatigue and corrosion (Mehrabi, 2006; Sun et al., 2013). Compared with other bridge components in the cable-stayed bridge, the cable is more vulnerable and usually has a shorter life. Therefore, in order to obtain the cable service state and ensure the safety of the bridge operation, cable monitoring has become an indispensable part of structural health monitoring (SHM) for cable-stayed bridges (Dong et al., 2018).

In cable-stayed bridges, the cable forces are usually considered as a significant safety indicator in monitoring process. Most of the cable monitoring approaches are to obtain the cable forces through various ways to evaluate the damage state of the cables, such as pressure gauge method, classical sensors method (Brice et al., 2008), vibration frequency method (Arjomandi et al., 2019; Cho et al., 2010; Huang et al., 2018; Ren et al., 2019; Sim et al., 2014), fiber bragg grating (FBG) method (Li et al., 2009), acoustic emission technology (Li et al., 2011; Zejli et al., 2012), vision-based monitoring method (Kim et al., 2013, 2017). Although these methods can obtain the cable tension forces, there are some limitations in practical applications. For instance, the method based on the cable fundamental vibration frequency is limited to long cables. The weak survivability and high cost of the smart cables with FBG-fiber make it not widely used. The accuracy of image-processing-based method is greatly affected by weather conditions.

In recent years, a few researchers utilize bridge deck strains, deck deflections, vertical dynamic displacements, deck shear forces, and rotation influence lines (RILs) to indirectly monitor the cable state. A method of detecting multiple damage in a cable-stayed bridge from the vertical dynamic of a vehicle crossing the bridge was presented by Yin and Tang (2011). It is difficult to measure the relative vehicle displacement in practice. The distributed strains along the bridge deck and the changes in support reactions are used by Nazarian et al. (2016a, 2016b) to detect the loss of cable forces, and then localize and quantify the damage in the cables. The laboratory and numerical results have been used to verify the proposed method. The partial cable damage using the abnormal variation of temperature-induced deck deflection caused by the cable damage was localized and quantified by Wang and Ye (2019). A cable damage identification technique based on the concept of RIL at the bridge bearing locations, and solely relies on measurements obtained from two points at either end of the bridge, for example, RIL_R and RIL_L was proposed by Alamdari et al. (2019). It is found that the cable damaged state will affect the bridge responses whether the bridge is under dead loads or live loads or temperature changes, and these indirect methods can identify the cable damage and bridge deck damage simultaneously. But most methods need to close traffic, which is difficult for bridges on traffic arteries. As the above, the operational environment has a significant effect on the identified results and it is limited the practical application.

Currently, with the development of artificial intelligence (AI), many researchers have introduced AI into structure damage identification or damage detection ways and have achieved many good results. Damage identification is formulated as an optimization problem, Ghannadi and Kourehli, 2019a; Ghannadi and Kourehli, 2019b; Ghannadi and Kourehli, 2020; Ghannadi et al., 2020) used Moth-Flame Optimization, Salp Swarm Algorithm, Multiverse Optimizer, and the grey wolf optimization to identify the damage severities and locations of space truss and steel frame. Ghannadi and Kourehli, 2019c; Ghannadi and Kourehli, 2021) also used artificial neural network (ANN) to detect the damage of skeletal structures. A two-step damage identification method combining a multilayer neural network and novelty detection is developed to robustly distinguish damage occurrence and severity regardless of temperature variations and noise perturbations by Gu et al. (2017). A graph neural network framework is trained by the Message Passing Neural Network to identify damaged cables and estimate the cable cross-sectional areas by Son et al. (2021). A fully convolutional network called Ci-Net for structural crack identification based on Pixel-level labeled image training data is proposed by Ye et al. (2019).

Among different AI techniques, support vector machine (SVM) is one of the youngest machine learning methods, which is proposed by Vapnik (1998). The most remarkable characterization of SVM is the high potential of generalizing datasets whose number is small in the training stage and SVM does not get stuck in local optimum like ANNs (Najafzadeh and Oliveto, 2020). So, SVM have ever applied in solving various problems, such as scour prediction (Najafzadeh et al., 2016), prediction of oxygen demand (Najafzadeh and Zeinolabedini, 2019), environmental evaluation (Najafzadeh and Zeinolabedini, 2019), face detection, character recognition, three-dimension body recognition, remote sensing image analysis, etc. For these advantages and widely application, SVM is also used in the field of damage detection and structural identification (Hasni et al., 2017). Least squares support vector machine and incomplete static responses of a damaged structure are used to detect structural damage by Kourehli (2017). SVM was employed as a classifier to evaluate the capability of feature selection techniques toward improving the damage identification performance of the cable-stayed bridge by Bisheh et al. (2020). SVM classifiers were developed to fuse the clustered features and identify multiple damage states by Hasni et al. (2017). A framework was developed for data-driven structural diagnosis and damage detection using SVM by Pan et al. (2018). PSO-SVM classification model was employed to automatically identify longitudinal crack, transverse crack, surface corrosion, and pothole defect by Li et al. (2020). The bridge deck’s deformation is mainly the bending deformation under moving vehicle loads, and the bending strains can be easily measured with less error using strain sensors. Furthermore, few researchers regard the problem of cable damage identification as a problem of classification and regression.

The aim of this study is to develop a novel method to estimate the cable state using SVM from the bending strain changes under moving vehicle loads. The analytical formulations between the damaged cable force and the bridge deck bending strains are established firstly using the finite element model (FEM). Then, a novel two-step cable damage identification approach is proposed: the damaged cable identification and the cable damage severity prediction. In the first step, a damage index vector (DIV) is defined. The damaged cable identification is considered as a multiple classification problem, and Support Vector Classification Machine (SVCM) is used to train the Damaged Cable Identification Machine (DCIM). In the second step, the damage severity identification of the cable is conducted and it is a nonlinear regression problem. Support Vector Regression Machine (SVRM) is utilized to train the Damage Severity Identification Machine (DSIM). The proposed method is applied to a field cable-stayed bridge to identify single damaged cable and double damaged cables to verify its feasibility and effectiveness.

Damage identification approach

Support vector machine

Support vector machine is one of the effective supervised machine learning techniques in data classification and regression (Pan et al., 2018). As shown in Figure 1, the key point of SVM is using the kernel function to transform the nonlinear problem in the original space into a linear problem in a higher dimensional feature space. Various kernel functions can be used, such as the linear, polynomial, or Gaussian RBF. In SVCM, a hyperplane is used to separate the two different classes of samples by maximizing the “margin,” which is the double distance from the hyperplane to the closest data points in either class (Pan et al., 2018), shown in Figure 1(a). In SVRM, a hyperplane is used to fit the function by minimizing ε, which is the distance of the hyperplane moving up and down along the vertical axis, and all training points are between the two dotted lines, shown in Figure 2(b). Detailed information is given in the book by Vapnik (1998).

Figure 1.

Support vector machine basic idea diagram. (a) Support vector classification machine. (b) Support vector regression machine

Figure 2.

The real cable-stayed bridge. (a) An illustration of the bridge. (b) The beam array (Alamdari et al., 2019). (c) Schematic view of the cable-stayed bridge (Kalhori et al., 2018).

In this study, the SVM toolbox in Matlab is used to train the cable damage identification machines. The following sections will describe the details.

The cable-stayed bridge

In this paper, a field cable-stayed bridge over the Great Western Highway in the state of New South Wales, Australia, was used as a case study to test and validate the proposed method. Figure 2(a) shows an illustration of the bridge (Alamdari et al., 2019). The cable-stayed bridge has a single A-shaped steel tower with a composite steel-concrete deck. The bridge is composed of 16 stay cables with a semi-fan arrangement. The bridge span and the tower height are 46 and 33 m, respectively. The deck has a thickness of 0.16 m and a width of 6.3 m, and it is supported by four I-beam steel decks. These decks are internally attached by a set of equally spaced floor beams, as depicted in Figure 2(b). The assigned numbers of the cables and the decks are shown in Figure 2(c).

The relationship between the cable force and the bending strains of the bridge deck

Based on the aforementioned real cable-stayed bridge in the state of New South Wales, a simple plane calculation diagram shown in Figure 3(a) is employed for deriving the equations between the cable forces and the bridge deck bending strains. In practice, the cable forces change with the vehicle location x when it is traveling on the bridge. To simplify the model, the cables can be removed and replaced by concentrated forces (Nazarian et al., 2016b), as shown in Figure 3(b). A roller support is used to represent the pylon to simplify the bridge as a continuous beam with two spans. Then the continuous beam is divided into six elements with seven nodes shown in Figure 3(b). The final calculation model is shown in Figure 3(c) with horizontal and vertical components of cable forces F_ih and F_iv as follows

F_{i h} = F_{i} \cos θ_{i} F_{i v} = F_{i} \sin θ_{i}

(1)

where θ_i is the angle between the deck and the ith cable.

Figure 3.

The cable-stayed bridge calculation diagram. (a) a simple plane calculation diagram (b) the diagram of the concentrated forces used to simulate the cable forces (c) the final calculation model.

The equilibrium equation can be obtained as follows

F (x) = K \cdot Δ

(2)

where F is the load vector

F (x) = {(\begin{array}{l} F_{1 h} (x) & F_{1 v} (x) & 0 & F_{2 h} (x) & F_{2 v} (x) & 0 & F_{3 h} (x) & F_{3 v} (x) & 0 & F_{4 h} (x) & F_{4 v} (x) & 0 & \dots \dots & 0 & 0 & 0 & 0 & 0 \end{array})}_{1 \times 17}^{T}

(3)

K is the beam stiffness matrix that has considered the boundary conditions

K = [\begin{array}{l} k_{1,1} & \dots & k_{1,17} \\ ⋮ & ⋱ & ⋮ \\ k_{17,1} & \dots & k_{17,17} \end{array}]

(4)

where

k_{i, j}

is the stiffness coefficient, which means the ith load when the jth displacement equal to 1 and other displacements are all equal to 0. Δ is the nodes displacements vector

Δ (x) = {(\begin{array}{l} u_{1} (x) & v_{1} (x) & φ_{1} (x) & u_{2} (x) & v_{2} (x) & φ_{2} (x) & u_{3} (x) & v_{3} (x) & φ_{3} (x) & \dots \dots & u_{4} (x) & v_{4} (x) & φ_{4} (x) & φ_{5} (x) & u_{6} (x) & φ_{6} (x) & u_{7} (x) & φ_{7} (x) \end{array})}_{1 \times 17}^{T}

(5)

where u_i, v_i and φ_i are the ith node horizontal displacement, vertical displacement, and rotation displacement, respectively.

From equation (2), Δ can be obtained as

Δ (x) = K^{- 1} \cdot F (x)

(6)

where

K^{- 1}

is the inverse matrix of K as follows

K^{- 1} = [\begin{array}{l} s_{1,1} & \dots & s_{1,17} \\ ⋮ & ⋱ & ⋮ \\ s_{17,1} & \dots & s_{17,17} \end{array}]

(7)

After obtaining the displacement of all nodes, each node’s moment and bending strain can be obtained by analyzing the element. Here, element is taken as an example to calculate the bending strain. The both ends force vector of element can be represented as

\bar{F} (x) = \bar{K} \cdot δ (x)

(8)

where

\bar{F} (x) = {(\begin{array}{l} \begin{array}{l} \begin{array}{l} F_{N i} (x) & F_{s i} (x) \end{array} & M_{i} (x) \end{array} & \begin{array}{l} \begin{array}{l} F_{N j} (x) & F_{s j} (x) \end{array} & M_{j} (x) \end{array} \end{array})}^{T}

(9)

where

F_{N i}

F_{N j}

F_{s i}

F_{s j}

and

M_{i}

M_{j}

are the axial forces, shearing forces and bending moments of the ith node and the jth node on element respectively.

The stiffness matrix of element is

\bar{K} = [\begin{array}{l} \begin{array}{l} k_{11} & k_{12} & k_{13} \\ k_{21} & k_{22} & k_{23} \\ k_{31} & k_{32} & k_{33} \end{array} & \begin{array}{l} k_{14} & k_{15} & k_{16} \\ k_{24} & k_{25} & k_{26} \\ k_{34} & k_{35} & k_{36} \end{array} \\ \begin{array}{l} k_{41} & k_{42} & k_{43} \\ k_{51} & k_{52} & k_{53} \\ k_{61} & k_{62} & k_{63} \end{array} & \begin{array}{l} k_{44} & k_{45} & k_{46} \\ k_{54} & k_{55} & k_{56} \\ k_{64} & k_{65} & k_{46} \end{array} \end{array}]

(10)

δ (x)

is the displacements of the ith node and the jth node on element

δ (x) = {(\begin{array}{l} \begin{array}{l} \begin{array}{l} u_{i} (x) & v_{i} (x) \end{array} & φ_{i} (x) \end{array} & \begin{array}{l} \begin{array}{l} u_{j} (x) & v_{j} (x) \end{array} & φ_{j} (x) \end{array} \end{array})}^{T}

(11)

Then the bending strain $ɛ_{i}$ can be obtained from the formula

ɛ_{i} (x) = \frac{M_{i} (x) y}{E I}

(12)

where y is the distance from the neutral axis to the outmost cross-section fiber. E is the modulus of elasticity of the deck. I is the moment of inertia of the deck cross-section.

Here, element $②$ is taken as an example to calculate the bending strains. If the local coordinate system is consistent with the global coordinate system, the nodes’ displacement vector of element $②$ is

δ {(x)}^{②} = {(\begin{array}{l} \begin{array}{l} \begin{array}{l} u_{3} (x) & v_{3} (x) \end{array} & φ_{3} (x) \end{array} & \begin{array}{l} \begin{array}{l} u_{2} (x) & v_{2} (x) \end{array} & φ_{2} (x) \end{array} \end{array}}^{T}

(13)

Using equations (3), (6) and (7), $δ {(x)}^{②}$ can be obtained as following

δ {(x)}^{②} = S^{'} F^{'} = [\begin{array}{l} \begin{array}{l} s_{71} & s_{72} & s_{74} \\ s_{81} & s_{82} & s_{84} \end{array} & \begin{array}{l} s_{75} & s_{77} & s_{78} \\ s_{85} & s_{87} & s_{88} \end{array} & \begin{array}{l} s_{7,10} & s_{7,11} \\ s_{8,10} & s_{8,11} \end{array} \\ \begin{array}{l} s_{91} & s_{92} & s_{94} \\ s_{41} & s_{42} & s_{44} \end{array} & \begin{array}{l} s_{95} & s_{97} & s_{98} \\ s_{45} & s_{47} & s_{48} \end{array} & \begin{array}{l} s_{9,10} & s_{9,11} \\ s_{4,10} & s_{4,11} \end{array} \\ \begin{array}{l} s_{51} & s_{52} & s_{54} \\ s_{61} & s_{62} & s_{64} \end{array} & \begin{array}{l} s_{55} & s_{57} & s_{58} \\ s_{65} & s_{67} & s_{68} \end{array} & \begin{array}{l} s_{5,10} & s_{5,11} \\ s_{6,10} & s_{6,11} \end{array} \end{array}] (\begin{array}{l} \begin{array}{l} F_{1 h} (x) \\ F_{1 v} (x) \\ F_{2 h} (x) \end{array} \\ \begin{array}{l} \begin{array}{l} F_{2 v} (x) \\ F_{3 h} (x) \end{array} \\ \begin{array}{l} F_{3 v} (x) \\ F_{4 h} (x) \end{array} \\ F_{4 v} (x) \end{array} \end{array})

(14)

where

F^{'}

is the cable force vector composed of non-zero forces,

S^{'}

is the corresponding sub-matrix of

K^{- 1}

Substituting equation (14) into equations (8) and (12), ε₃ and ε₂ can be obtained as

(\begin{array}{l} ɛ_{3} (x) \\ ɛ_{2} (x) \end{array}) = (\begin{array}{l} \frac{M_{3} (x) y}{E I} \\ \frac{M_{2} (x) y}{E I} \end{array}) = B F^{'} = (\begin{array}{l} \begin{array}{l} B_{11} \\ B_{21} \end{array} & \begin{array}{l} \begin{array}{l} B_{12} & B_{13} \end{array} \\ \begin{array}{l} B_{22} & B_{23} \end{array} \end{array} & \begin{array}{l} \begin{array}{l} B_{14} \\ B_{24} \end{array} & \begin{array}{l} \begin{array}{l} B_{15} & \begin{array}{l} B_{16} & B_{17} & B_{18} \end{array} \end{array} \\ \begin{array}{l} B_{25} & \begin{array}{l} B_{26} & B_{27} & B_{28} \end{array} \end{array} \end{array} \end{array} \end{array}) (\begin{array}{l} \begin{array}{l} F_{1 h} (x) \\ F_{1 v} (x) \\ F_{2 h} (x) \end{array} \\ \begin{array}{l} \begin{array}{l} F_{2 v} (x) \\ F_{3 h} (x) \end{array} \\ \begin{array}{l} F_{3 v} (x) \\ F_{4 h} (x) \end{array} \\ F_{4 v} (x) \end{array} \end{array})

(15)

where

B = \frac{y}{E I} (\begin{array}{l} \begin{array}{l} k_{31} \\ k_{61} \end{array} & \begin{array}{l} \begin{array}{l} k_{32} & k_{33} \end{array} \\ \begin{array}{l} k_{62} & k_{63} \end{array} \end{array} & \begin{array}{l} \begin{array}{l} k_{34} \\ k_{64} \end{array} & \begin{array}{l} \begin{array}{l} k_{35} & k_{36} \end{array} \\ \begin{array}{l} k_{65} & k_{66} \end{array} \end{array} \end{array} \end{array}) [\begin{array}{l} \begin{array}{l} s_{71} & s_{72} & s_{74} \\ s_{81} & s_{82} & s_{84} \end{array} & \begin{array}{l} s_{75} & s_{77} & s_{78} \\ s_{85} & s_{87} & s_{88} \end{array} & \begin{array}{l} s_{7,10} & s_{7,11} \\ s_{8,10} & s_{8,11} \end{array} \\ \begin{array}{l} s_{91} & s_{92} & s_{94} \\ s_{41} & s_{42} & s_{44} \end{array} & \begin{array}{l} s_{95} & s_{97} & s_{98} \\ s_{45} & s_{47} & s_{48} \end{array} & \begin{array}{l} s_{9,10} & s_{9,11} \\ s_{4,10} & s_{4,11} \end{array} \\ \begin{array}{l} s_{51} & s_{52} & s_{54} \\ s_{61} & s_{62} & s_{64} \end{array} & \begin{array}{l} s_{55} & s_{57} & s_{58} \\ s_{65} & s_{67} & s_{68} \end{array} & \begin{array}{l} s_{5,10} & s_{5,11} \\ s_{6,10} & s_{6,11} \end{array} \end{array}]

(16)

Equation (15) shows the relationship between cable forces $F^{'}$ and the bending strains ε₃ and ε₂ of the bridge deck. From this equation, it can be found that each bending strain will change with the cable forces. This conclusion also applies to the three-dimensional model, but the expression is more complicated. It is well known that the cable forces are varied with different vehicle loads and different vehicle locations, and the bending strains of the bridge with the crossing vehicle are time-varying. When one of the cables is damaged, all cable forces will be redistributed and the corresponding bending strains can reflect the damage of the cables.

Damage identification approach

Damage in cables as a result of corrosion changes the cable cross-section and potentially mass per unit length (Huang et al., 2018). The cable damage will affect the cable tensile stiffness EA, furthermore affect the cable tension forces. Therefore, in this paper, the cable damage is described using the cable tensile stiffness EA decreasing, and the damage severity is defined as

d e = \frac{E A_{in} - E A_{da}}{E A_{in}} \times 100 %

(17)

where de is the cable damage severity, EA_in is the intact cable tensile stiffness, and EA_da is the damaged cable tensile stiffness.

From the previous section, the bridge deck bending strains can reflect the damage of the cables and are able to be measured easily with good accuracy. The DIV will be extracted from measured bending strains. However, the amount of strain data is very large because of the variation with vehicle location. In order to reduce the large amount of data, the maximum bending strains of the measured points are extracted as the main features of the data. For a beam structure, the position where the concentrated force act is usually the extreme point of the bending moment and also the extreme point of the bending strain. In this study, the strain measurement points on the bridge deck are corresponding to eight cable anchors of the cable-stayed bridge. Then the damage identification index for each measurement point is defined as the difference between the maximum intact bending strain and the maximum damaged bending strain. DIV {X} is obtained

{X} = {(\begin{array}{l} \begin{array}{l} Δ ɛ_{1} & \dots \end{array} & \begin{array}{l} Δ ɛ_{i} & \dots \end{array} & Δ ɛ_{n} \end{array})}^{T}

(18)

where

Δ ɛ_{i}

is the vector component at the ith measured point, and the index is calculated by

Δ ɛ_{i} = \max (ɛ_{i in}) - \max (ɛ_{i da})

(19)

where

\max (ɛ_{i in})

and

\max (ɛ_{i da})

are the maximum bending strains of intact bridge and damaged bridge at the ith measured point under the same vehicle load, respectively.

The cables are numbered and the damaged cable identification can be defined as a multi-class classification problem. It means that the number of categories will be nc, when the total number of the cables on cable-stayed bridge is nc. SVCM is a Multiple Classification Methods (MCM) and it can be used to train a DCIM, as the following function

DCIM ({X}) = \arg max {\begin{cases} sgn (f_{1} ({X})), & \dots, & sgn (f_{n c} ({X})) \end{cases}}

(20)

where

{X}

is DIV. sgn (f_nc (X)) is a two-class classifier, when

{X}

is corresponding to the damaged cable of nc,

sgn (f_{n c} ({X})) = 1

, otherwise,

sgn (f_{n c} ({X})) = - 1

. DCIM

({X})

equals the number of the damaged cable which is corresponding to the maximum of the two-class classifiers. So, DCIM

({X})

is a multi-class classifier. Consequently, the damaged cable can be identified by substituting any DIV into equation (20).

After determining the damaged cable, the next step is to identify the cable damage severity. The DSIM will be trained using the damage severity identification index datasets which only includes the data of one damaged cable situation. Corresponding to the number of the bridge cables, the number of DSIMs should be nc, here it is 16. Because the damage severity is a continuous variable, it is impossible to prepare the training data corresponding all of the damage severity. The damage severities such as 5%, 10%, 15%, and 20%, etc. are used for the training data. Then the regression method, SVRM, is utilized to train DSIMs. In DSIM, DIV ${X}$ is the independent variable and the damage severity de is the dependent variable. DSIM is a nonlinear higher dimensional function as

d e = {DSIM}_{i c} ({X}) = {A} \cdot {X} + b

(21)

where DSIM_ic ({X}) is for damaged cable ic,

{A} = (\begin{array}{l} a_{1} & \begin{array}{l} a_{2} & \dots \end{array} & a_{n} \end{array})

is weight coefficient row vector, b is constant. Substituting DIV into equation (21), the result de is the damage severity.

In summary, this cable damage identification approach should be separated into two identification machines, DCIM and DSIM. The flow chart of the damaged cable identification is shown in Figure 4.

Figure 4.

The flow chart of the damaged cable identification.

The structural health monitoring system and finite element model of the cable-stayed bridge

Description of the structural health monitoring system

A long-term monitoring system has been installed on a cable-stayed bridge as shown in Figure 5. There are 24 accelerometers on the bridge deck and their locations are indicated in Figure 5(a). The locations of the strain gauges are elaborated in Figure 5(b). All eight cables were instrumented with uni-axial strain gauges, denoted by SAi (i = 23–30) in Figure 5(b). Uni-axial strain gauges SUi (i = 17–22) were mounted under the deck in either the longitudinal or transverse direction between Cross Girders CG6 and CG7. Strain gauges SUi (i = 13–16) were installed under the flange of the longitudinal girders at the middle of the span between CG6 and CG7 to measure bending strains. These strain gauges were also located close to mid-span of the bridge, where large deflections are expected. Shear rosettes were mounted at three different longitudinal locations along the bridge: the north end of the span near Cross Girder CG2, bridge mid-span close to Cross Girder CG6, mid-span close to Cross Girder CG6, and halfway between Cross Girders CG6 and CG7 (Alamdari et al., 2019). These strain gauges are implemented to collect shear, bending, and tension strains to characterize passing traffics and to identify the vehicle gross and axle weights. The response signals of the bridge were collected at 600 Hz.

Figure 5.

Illustration of the cable-stayed bridge structural health monitoring. (a) Illustration of the accelerometer sensor locations (A1:A24) on the cross girders (CGs) (Kalhori et al., 2018). (b) Illustration of the strain gauges array (Kalhori et al., 2018).

Finite element model of the cable-stayed bridge

The finite element model of the cable-stayed bridge was established utilizing ANSYS. The bridge deck is a composite steel-concrete deck, so the steel reinforced concrete part is simulated by SHELL63 with 160 mm thickness, the under longitudinal and transverse girders are simulated by BEAM189 which have Universal Beam (410UB54) cross-sectional properties. These two kinds of elements are coupled at their co-nodes. The cables are super grade circular bar with a diameter of 38 mm and are all pre-tensioned cables using LINK10 to simulate. At the cable anchorage footing, all of the severities of freedoms are restrained. The bridge mast has a non-prismatic cross-section, starting with a rectangular section of 700 mm × 800 mm at the base and 500 mm × 800 mm at the uppermost level. Consequently, in the cable-stayed bridge FEM, BEAM189 with variable section is adopted to simulate the mast, and all severities of freedoms of the mast base are restrained. At abutment A (in Figure 2(c)), there is a pin support under every longitudinal deck. On the other side of the deck, roller supports are used at abutment B (in Figure 2(c)). The FEM of the cable-stayed bridge is shown in Figure 6.

Figure 6.

The finite element model of the cable-stayed bridge under vehicle loads.

Verification of the finite element model

Based on the SHM system monitoring dynamic data, the FEM of the cable-stayed bridge is updated. In the FEM, some parameters are uncertain. For instance, because the cross girders have many circular openings for service ducts (in Figure 2(b)), their cross-section areas and moment of inertia are uncertain. And because the deck has reinforced steels and pavement, its density and modulus of elasticity are uncertain. An objective function can be established as

f_{1} = FEM (A_{c}, I_{c}, E_{d})

(22)

where f₁ is the monitoring first natural frequency. FEM () is the finite element model of the cable-stayed bridge. A_c and I_c are the updated cross-section areas and moment of inertia of the cross girder, respectively. E_d is the updated modulus of elasticity of the deck. Since A_c and I_c are related to the cross-girder tension stiffness E_cA_c and bending stiffness E_cI_c, its cross-section can take the cross-section of Universal Beam (410UB54), and only the modulus of elasticity E_c is modified. The objective function is simplified as

f_{1} = FEM (E_{c}, E_{d})

(23)

Here, E_d=43.2 GPa, E_c=210 GPa.

Then, the FEM modals and frequencies are calculated and compared with the test data. Table 1 shows the first five FEM frequencies and the test frequencies (Sun et al., 2017). Figure 7 shows the first test mode shape and the first five mode shapes of the cable-stayed bridge.

Table 1.

Frequencies from the finite element model and filed measurements.

Mode no	FEM frequency (Hz)	Test frequency (Hz) (Sun et al., 2017)	The difference (Hz)
1	2.038	2.014	0.024
2	3.163	3.510	−0.347
3	4.088	3.645	0.443
4	5.329	5.538	−0.209
5	6.530	6.068	−0.462

Note: The difference = FEM Frequency- Test Frequency.

FEM: finite element model.

Figure 7.

The mode shapes of the cable-stayed bridge. (a) The 1st test mode shape (Zhu et al., 2017). (b) The 1st finite element model mode shape at side view. (c) The first five modes of 3D perspective.

From Table 1, it can be found that the corresponding frequencies are close, and the maximum difference is −0.462 Hz, which is the fifth frequency. Figure 7 shows that the first FEM mode shape and the corresponding test mode shape (Zhu et al., 2017) are agreed well, and the mode’s modal assurance criterion (MAC) is 0.9778, which is calculated by (Ghannadi and Kourehli, 2020a)

{MAC}_{i} = \frac{{| {\emptyset_{i}^{c}}^{T} {\emptyset_{i}^{m}} |}^{2}}{({\emptyset_{i}^{c}}^{T} {\emptyset_{i}^{c}}) ({\emptyset_{i}^{c}}^{T} {\emptyset_{i}^{c}})}

(24)

where

{\emptyset_{i}^{c}}

and

{\emptyset_{i}^{m}}

denote the ith calculated mode shape and measured mode shape, respectively. When MAC_i is closer to 1,

{\emptyset_{i}^{c}}

would fit well with

{\emptyset_{i}^{m}}

From these results, it is indicated that the FEM can represent the bridge global stiffness.

Furthermore, because this paper mainly uses the bending strain to obtain the damage identification indexes, the calculated bending strain is also compared with the measured bending strain to verify the FEM’s correctness. The response measuring sensor is the SU15, which is located on the longitudinal deck just in between transverse beam CG6 and CG7 as indicated in Figure 5(b). In the field test experiment, a Holden Colorado Ute was used as a test vehicle, see Figure 8(a). The gross weight of the test vehicle was 2.2 t, with front and rear axle weights of P1 = 1.20 t and P2 = 1.00 t, respectively. The distance between the axles was d = 3.1 m (Alamdari et al., 2019). The test vehicle was driving at approximately a constant speed of 10 km/h from the South to the North of the bridge along the centerline. Figure.8(b) illustrates the measured strain response (black line) and the FEM strain response (red line). The presence of two peaks, corresponding to two axles with front axle being heavier than the rear axle, can be clearly seen in both the measured and calculated bending responses in Figure 8(b). And these two lines are very close, the maximum values are almost the same. From the above, the FEM is validated.

Figure 8.

The test vehicle and the corresponding strains. (a) Holden Colorado Ute (Alamdari et al., 2019). (b) The measured strain response (black line) (Alamdari et al., 2019) and the finite element model strain response (red line).

Single cable damage identification

The damage identification indexes

The damage identification indexes are obtained from the bending strain responses of the longitude deck which are near the cable anchor. The vehicle loads also used the Holden Colorado Ute which travels along the bridge centerline. Figure 9 shows the bending strain response under the cable 1–4 and cable 9–12 (Figure 2(c)), when the bridge is intact and the cable 4 is damaged 30%.

Figure 9.

Bending strain response under the cable 1–4 and cable 9–12, when the bridge is intact and the cable 4 is damaged 30%.

From Figure 9, it is obviously that the deck bending strain would be changed when the cable is damaged, especially near the damaged cable, such as the cable 4. But only according to these curve lines cannot identify the damaged cable or the damage severity. From Figure 9, for example, it is difficult to determine which cable is damaged, the cable 4 or the cable 3. Therefore, the new damaged identification approach in the Relationship Between the Cable Force and the Bending Strains of the Bridge Deck section can be used to identify the damaged cable and its damage severity.

According the Relationship Between the Cable Force and the Bending Strains of the Bridge Deck section, corresponding to each cable anchor (cable 1–4, and cable 9–12), the difference between the maximum bending strains can be calculated by equation (19) to obtain a DIV, including 8 differences. Figure 10 shows the DIVs poly line diagrams when the cable 1–8 cross sections are reduced 30%, respectively. The measured points 1–8 are, respectively, under the cable anchors of cable 1–4 and cable 9–12. It indicates that the poly line diagrams are significant unique from each other when the cable 1–4 are damaged, respectively. The poly line diagrams shapes of the cable 5–8 are similar, but each corresponding measured point values are different.

Figure 10.

The damage indexes vector poly line diagrams when the cable 1–8 cross sections are reduced 30%.

Furthermore, using FEM, when damage severity of the cable 4 are 10%, 20%, and 30%, respectively, the corresponding DIV are calculated and their poly line diagrams are shown in Figure 11. It shows that the absolute values of damage identification index increase as the damage severity increases in cable 4, and their shapes are similar. Therefore, the cable damage severity identification can be considered as a regression problem.

Figure 11.

The damage indexes vector poly line diagrams when the cable 4 cross-section is reduced 10%, 20% and 30%.

Training datasets and testing datasets

Before training DCIM and DSIM, the training datasets and the testing datasets should be prepared firstly.

1. Training datasets

For the damaged cable identification, the original training datasets {x}^ol contains 16 DIVs, when the cable 1–16 are damaged 30%, respectively. So, the corresponding damaged cable labels vector is {y}^ol = (1 2 3 … 16)^T. In order to increase the number and the randomness of the training datasets, there are three steps to process the original datasets. First of all, the original datasets are expanded by equation (25) (Ren et al., 2013).

{\begin{cases} {x}_{n \times 16}^{e l} = {x}_{j}^{o l} \times (I_{n \times n} + ɛ \cdot diag {(R)}_{n \times n}) \\ {y}_{n \times 1}^{e l} = y_{j}^{o l} \times {ones}_{n \times 1} \end{cases}

(25)

where

{x}_{n \times 16}^{e l}

and

{y}_{n \times 1}^{e l}

are the expanded data vector of the jth original DIV and the corresponding damaged cable label vector, respectively.

{x}_{j}^{o l}

and

y_{j}^{o l}

are the jth original DIV and the corresponding damaged cable label, respectively.

I_{n \times n}

is the identity matrix. ε is the noise level.

diag {(R)}_{n \times n}

is a diagonal matrix, which diagonal component is a normally distributed random data with a mean of 0 and a mean square deviation of 1.

{ones}_{n \times 1}

is a vector with all components being 1. Here, n is 15 and ε is 0.1%, the number of original data is 16. After expanding, the number of samples is 240.

The second step is to add white noise into each component of the expanded training datasets by equation (26).

x_{i j}^{t l} = x_{i j}^{e l} \times (1 + ɛ R_{j}) (j = 1,2, \dots, m)

(26)

where

x_{i j}^{t l}

and

x_{i j}^{e l}

are the jth component of the ith training DIV and the jth component of the ith expanded vector. ε and R_i are same with the meaning in equation (25). Here, ε is 1% and m is 8 for the number of the measured points being 8.

At last, the training vectors should be normalized in [0,1] to increase the identification accuracy and reduce errors. In Matlab, it can be processed by the function “mapminmax,” and its algorithm is

f : x_{i j}^{t l} \to X_{i j}^{t r l} = \frac{x_{i j}^{t l} - x_{i j min}^{t l}}{x_{i j max}^{t l} - x_{i j min}^{t l}}

(27)

where

x_{i j min}^{t l}

and

x_{i j max}^{t l}

are the minimum component and the maximum component of the vector

x_{i}^{t l}

, respectively.

X_{i j}^{t r l}

is the normalized component. Then, the training datasets [X]^trl for the damaged cable identification is obtained, which is a 240

\times

8 matrix.

For damage severity identification, the cable 2 is taken as an example. The DIV are calculated using the former FEM, when the cable 2 damage severities are 10%, 20%, and 30%, respectively. The original damage severity identification training datasets {x}^od only contains 3 DIVs and their poly line diagrams have shown in Figure 12. The damage severity vector is {y}^od = (10 20 30)^T. They also are added white noise to simulate measured (SM) datasets. In equation (25), n is 40 and ε is 1%, and the number of the expanded training datasets is 160. In equation (26), ε is 1% and m is 8. It must be mentioned that this training datasets do not need to be normalized. Finally, the number of the training samples [x]^trd and the corresponding {y}^trd are 160.

2. Simulated measured datasets

Figure 12.

The comparison between the actual damaged cable and the identified damaged cable with different noise levels. (a) noise level is 0%, (b) noise level is 5%,(c) noise level is 10%,(d) noise level is 15%.

In order to test DCIM and DSIM generalization capability and anti-noise capability, the hypothetical damage scenarios are listed in Table 2 to SM datasets. DIVs in Table 2 are calculated by FEM.

Table 2.

The hypothetical damage scenarios.

	Damaged cable no	Damage severity	Remarks	Damaged cable no	Damage severity	Remarks
Damaged cable identification test scenarios	1	10%, 20%	Not included in the training datasets	10	25%	Not included in the training datasets
	2	10%, 20%, 25%		11	20%
	3	10%, 20%		14	20%
	4	10%, 20%
Damage severity identification test scenarios	2	10%, 20%, 25%, 30%	25% is not included in the training datasets

Then the damaged cable identification SM input dataset [X]^tel includes 12 damage scenarios. The exact output vector is {y}^tel = (1 1 2 2 2 3 3 4 4 10 11 14)^T. The damage severity identification SM test input datasets [x]^ted includes 4 damage scenarios, and output is {y}^ted = (10 20 30 25)^T. Then they are expanded by equations (25) to 80 damage scenarios, where n is 20 and ε is 0.01%.

At the following sections, the SVM will be utilized to train DCIM and DSIM, which are based on the above training datasets.

Damaged cable identification

It has been mentioned that the damaged cable identification is a multi-classification problem. And on the Matlab platform, the function “fitcecoc” fit multiclass models for SVCM. Therefore, this paper uses the function “fitcecoc” to train DCIM. The major training steps are as follows:

1. [X]t^rl are randomly divided into two subsets: 90% of [X]t^rl is used for training DCIM and 10% of [X]t^rl is used for testing DCIM.

2. Choosing the Gaussian kernel as the kernel function.

3. Using function “fitcecoc” to train DCIM.

4. Using function “crossva” to cross-validate DCIM using 10-fold cross-validation.

5. Using function “kfoldLoss” to get classification loss for cross-validated DCIM, it is 4% here.

Then [X]^tel are input in DCIM to get identified damaged cable label {y}^pl. {y}^pl and the actual damaged cables {y}^tel are shown in Figure 12(a). Although [X]^tel is not included in the training datasets, it can be observed from Figure 12(a) that the identified results are exactly correct. Since there are many disturbances in the field experiment environment, studying the anti-noise capability of DCIM is necessary. The following noise levels 5%, 10%, and 20% are considered and added into [X]^tel using equation (26). These noised SM datasets are input in DCIM and the identification results are show in Figure 12(b)–(d).

From Figure 12, even the noise level is up to 20%, there are not any misidentified damaged cables. Obviously, DCIM has excellent anti-noise capability and good generalization capability, and the DIV are also effective to identify the damaged cables.

Damage severity identification

After finishing the damaged cable identification, damage severity recognition will be studied in this part. At first, [x]^trd, which is obtained at the Training Datasets and Testing Datasets section, is used to train DSIM by SVRM at Matlab platform. The major training steps are as follows:

1. [x]t^rd and {y}t^rd are randomly divided into two subsets according the proportion 90% and 10%, [x]90 and {y}90 are used for training DSIM, [x]10 and {y}10 are used for testing DSIM.

2. Using function “fitrsvm” to train DSIM by optimizing hyperparameters automatically, such as:

“DSIM = fitrsvm([x] ₉₀ , {y} ₉₀ , 'OptimizeHyperparameters', ‘auto’, 'HyperparameterOptimizationOptions', struct(‘AcquisitionFunctionName’, ‘expected-improvement-plus'))”

3. [x]10 is input DSIM to get {y}^p, the codes are:

{y}^p = predict (DSIM, [x]₁₀)

4. The performance of DSIM is measured by mean squared error (MSE) (Ghannadi and Kourehli, 2021), regression correlation coefficient squared (R2), uncertainty interval U95 and Reliability (Sabed-Movahed, F et al., 2020). The mathematical relations of these parameters are given in equation (28)–(31).

M S E = \frac{1}{N} {\sum_{i = 1}^{N} (P_{i} - D_{i})}^{2}

(28)

R^{2} = \frac{\sum_{i = 1}^{N} P_{i}^{2}}{\sum_{i = 1}^{N} P_{i}^{2} - \sum_{i = 1}^{N} {(P_{i} - D_{i})}^{2}}

(29)

U 95 = \frac{1.96}{N} \sqrt{{\sum_{i = 1}^{N} (D_{i} - \bar{D})}^{2} + {\sum_{i = 1}^{N} (D_{i} - P_{i})}^{2}}

(30)

Reliability = (\frac{100 %}{N}) \sum_{i = 1}^{N} k_{i}

(31)

where P_i is the identified damage severity, D_i is the exact damage severity, N is the number of samples,

\bar{D}

is the mean value of the exact damage severity. k_i is obtained through two steps. First, the relative average error (RAE) is defined as a vector whose ith component is

R A E_{i} = | \frac{D_{i} - P_{i}}{D_{i}} |

(32)

Next is RAE_i ≤ Δ, then k_i = 1, otherwise, k_i = 0, where Δ is the threshold value, it can be 0.1 or 0.05.

When the MSE is closer to 0, R² is closer to 1, U95 is closer to 0 and Reliability is closer to 100, DSIM would have a good performance. Here, MSE is 0.3468, R² is 1.0008, U95 = 0.1490, Reliability is 100 when Δ = 0.1, or is 86.6667 when Δ = 0.05.

Then [x]^ted is input in the DSIM, and the output results are the damage severity of cable 2. Table 3 lists MSE, R², U95, and Reliability when the test data is added in different noise levels, such as 0%, 5%, 10%, 15%, and 20%.

Table 3.

The performance results of damage severity identification machine for mean squared error, R², U95, and Reliability.

	Noise level	0%	5%	10%	15%	20%	Remarks
Damage severity of 10%	MSE	0.1787	0.1834	0.3640	0.3964	0.6550	Included in the training datasets
	R ²	1.0016	1.0017	1.0033	1.0037	1.0065
	U95	0.1853	0.1877	0.2644	0.2760	0.3547
	Reliability (Δ = 0.1) (%)	100	100	90.00	95.00	80.00
Damage severity of 20%	MSE	0.2357	0.4677	0.8801	2.3850	2.1926
	R ²	1.0006	1.0012	1.0023	1.0066	1.0057
	U95	0.2128	0.2997	0.4122	0.6768	0.6490
	Reliability (Δ = 0.1) (%)	100	100	90.00	80.00	80.00
Damage severity of 30%	MSE	0.0679	0.6401	1.5673	6.1620	8.4525
	R ²	1.0001	1.0007	1.0017	1.0070	1.0090
	U95	0.1142	0.3506	0.5487	1.0879	1.2742
	Reliability (Δ = 0.1) (%)	100	100	100	90.00	65.00
Damage severity of 25%	MSE	0.1199	0.4228	1.3951	3.0524	5.0408	Not included in the training datasets
	R ²	1.0002	1.0007	1.0023	1.0052	1.0085
	U95	0.1517	0.2850	0.5177	0.7657	0.9840
	Reliability (Δ = 0.1) (%)	100	100	95.00	80.00	60.00

MSE: mean squared error.

From Table 3, with the noise level increasing, MSE, R², and U95 have a tendency to increase, but Reliability is decreasing. It indicates that DSIM performance is getting worse with the noise lever increasing. However, most of MSE are smaller than 6, except the situation of the damage severity of 30% with noise level 15% and 20%. All of R² are very close to 1, the maximum is only 1.0090. U95 is also smaller than 1.2742. When the noise level is less than 15%, the reliabilities are all above 80%. When the noise level is 20%, the reliabilities are also 80% of the damage severity of 10% and 20%. Although the damage severity of 25% is not included in the training datasets, DSIM still have good performance. Therefore, DSIM can well identify the damage severity, have good generalization capability, and strong anti-noise performance. And these DIV, which are proposed in this paper, are effective to identify the cable damage severities.

Double damaged cables identification

In real situation, there may be two damaged cables simultaneously. The key steps are also to train the double damaged cable identification machine (DDCIM) and the double damage severity identification machine (DDSIM). The double cables DIVs are obtained by superimposed the single cable DIVs corresponding different damaged cables. In DDCIM, the categories include all the scenarios when any two cables are damaged simultaneously. Such as, when the total number of cables are nn in a cable-stayed bridge, the categories are $\sum_{i = 1}^{n n - 1} i$ , which is more than that in DCIM. If the double damaged cables are identified, DDSIM needs to train for each cable damage severity separately.

The double damaged cables identification datasets

1. Training datasets

For double damaged cables identification, the double cables DIVs can be obtained as equation (33)

{\begin{cases} {x}_{k}^{d o l} = {x}_{i}^{o l} + {x}_{j}^{o l} \\ {y}_{k}^{d o l} = ({y}_{i}^{o l} {y}_{j}^{o l}) \end{cases} (i \neq j, i = 1,2, \dots, n n; j = 1,2, \dots, n n; k = 1,2, \dots, \sum_{i = 1}^{n n - 1} i)

(33)

where

{x}_{k}^{d o l}

and

{y}_{k}^{d o l}

are the kth DIV of the double damaged cables identification training datasets and the corresponding damaged cable labels, respectively.

{x}_{i}^{o l}

and

{x}_{j}^{o l}

are, respectively, the ith and the jth single cable DIV, and

{y}_{i}^{o l}

and

{y}_{j}^{o l}

are, respectively, the ith and the jth single damaged cable label, where i and j cannot equal to each other. nn is the total number of cables in the cable-stayed bridge, here nn = 16. It also can be expanded and added noise by equations (25) and (26).

For double damaged cables damage severity identification, different DSIMs should be trained for different damaged cables. The training datasets can be obtained as equation (34)

\begin{array}{l} {\begin{cases} {x}_{k}^{d o d} = {x_{c n}}_{i}^{o d} + {x_{c m}}_{j}^{o d} \\ {y}_{k}^{d o d} = ({y_{c n}}_{i}^{o d} {y_{c m}}_{j}^{o d}) \end{cases} & (i = 1,2, \dots, m m; j = 1,2, \dots, m m; k = 1,2, \dots, m m \times m m) \end{array}

(34)

where

{x}_{k}^{d o d}

and

{y}_{k}^{d o d}

are the kth DIV of the double damaged cables damage severity identification training datasets and the corresponding damage severities.

{x_{c n}}_{i}^{o d}

{x_{c m}}_{j}^{o d}

and

{y_{c n}}_{i}^{o d}

{y_{c m}}_{j}^{o d}

are the ith and the jth single cable DIV and damage severity for cable cn and cable cm, respectively. mm is the number of the single damage severity scenarios, such as 10%, 20%, 30%, mm = 3. They also can be added white noise to SM datasets by equations (25) and (26).

2. Testing datasets

In order to test the identification results of DDCIM and DDSIM, there are two hypothetical damage scenarios. In the scenario ①, cable 1 is damaged 10% and cable 2 is simultaneously damaged 20%. In the scenario ②, cable 1 is damaged 20% and cable 11 is simultaneously damaged 10%. The corresponding DIV are directly calculated by FEM and are added white noise by equation (26) to get the testing datasets DATA_td, the noise levels include 5%, 10%, and 20%. For DDCIM, the actual output is ${y}_{t e s t}^{d o l} = (\begin{array}{l} 1 & 2 \\ 1 & 11 \end{array})$ . For DDSIM, the actual output is ${y}_{t e s t}^{d o d} = (\begin{array}{l} 10 & 20 \\ 20 & 10 \end{array})$ .

The double damaged cables identification

Double damaged cable identification machine is trained on the Matlab platform, and the training steps are same as the Damaged Cable Identification section. Then, the testing datasets DATA_td are input into DDCIM, the output ${y}_{i d}^{d o l} = (\begin{array}{l} 1 & 2 \\ 1 & 11 \end{array})$ is completely correct.

Although the training datasets are just obtained by superposition principle, DDCIM can exactly identify the damaged cables, even when the noise level is 20%. It is shown that the training datasets can be obtained by superposition principle based on single cable damage datasets for multiple damage scenarios, such as two cables damage or more. It also indicates that DDCIM has good anti-noise capability.

Damage severity identification

If damage scenario ② has been identified, the training datasets

{x}^{d o d}

can be obtained by equation (34), where cn=1, cm=11. DDSIM is trained by SVRM on Matlab platform, and the training steps can be found in the Damage Severity Identification section. Then the testing DIV datasets about damage scenario ② is input into DDSIM₁ and DDSIM₁₁, respectively. The identified damage severity, the exact damage severity with different noise levels and the errors are all shown in Table 4.

Table 4.

The double damaged cables damage severity identification results of damage scenarios ②.

Noise level, %	The assumed damaged cable	The exact damage severity (%)	The identified damage severity (%)	Error (%)
0	1	20	19.21	−0.79
0	11	10	10.08	0.08
5	1	20	18.90	−1.10
5	11	10	10.29	0.29
10	1	20	18.18	−1.82
10	11	10	10.66	0.66
20	1	20	20.25	0.25
20	11	10	8.86	−1.14

where error = identified damage severity – exact damage severity.

From Table 4, the identified damage severities are very close to the exact damage severities, the maximum error is just −1.82% of the damaged cable 1 with the noise level of 10%. When the noise level is lower than 20%, the identified error is less than 2%. This example also shows that the method of obtaining the training datasets using superposition principle based on single cable damage datasets for multiple damage condition is correct and can be used in real condition. DDSIM has good anti-noise capability.

Conclusions

Bending strain-based DCIM and DSIM have been developed to identify the damaged cable in a cable-stayed bridge. A cable-stayed bridge FEM is established using ANSYS, and validated by the field measurements. The validated FEM is used to generate the training and testing datasets for DCIM and DSIM. Some conclusions can be made as follows:

1. DCIM can accurately identify the single damaged cable from the bending strain measurements with the noise level up to 20%. DSIM works well on identifying the damage severities, the maximum error is 4.9329% when the noise level is 20%. The identification results show that both DCIM and DSIM have the good generalization capability and anti-noise capability.

2. Based on the single damaged cable identification datasets, DDCIM and DDSIM are developed to identify double damaged cables. The results show that both DDCIM and DDSIM are accurate and robust to identify the double damaged cables. The identified error is less than 2% when the noise level is up to 20%.

3. The proposed method has strong anti-noise performance and can be easily adapted to the field health monitoring system. The scenarios with three or more damaged cables are much complicated to be discussed next step. The effect of the geometric nonlinearity and material nonlinearity need to be considered for practical application in the long-span cable-stayed bridges. Further study is needed to develop an integrating system for simultaneously identifying the damage in both the cable and bridge deck.

Footnotes

Acknowledgement

Figures 2, 5, and 8 reprinted from Engineering Structures, vol. 185, Mehrisadat Makki Alamdari, Kamyar Kildashti, Bijan Samali, Hamid Valipour Goudarzi, “Damage diagnosis in bridge structures using rotation influence line: Validation on a cable-stayed bridge”, pp 1-14, 2019 with permission from Elsevier.

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by research funding of National Natural Science Foundation of China (NSFC) (11972238, 11902206, 51878433) and the Australian Research Council Discovery Project (DP160103197). The financial aid is gratefully acknowledged.

Correction (November 2023):

Article updated to correct Figures 2,5,8.

ORCID iDs

Jianying Ren

Xinqun Zhu

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