## 3 Case Study  
### Dataset  
Cables are one of the most critical and vulnerable components in a cable-stayed bridge that suffer cyclic loads and harsh environments [49; 50], and cable tension force is the most direct indicator. Opensource cable tension dataset of an in-service cable-stayed bridge released in 1 st IPC-SHM (official website @ [[http://www.schm.org.cn/#/IPC-SHM,2020](http://www.schm.org.cn/#/IPC-SHM,2020)]([http://www.schm.org.cn/#/IPC-SHM,2020](http://www.schm.org.cn/#/IPC-SHM,2020)) /dataDownload, (accessed on 15 January 2021)) is used for the case study. As shown in Figure 4, the bridge is a double-cable-plane cable-stayed bridge with 168 cables (84 pairs). All cables are allocated load cells for the monitoring of the dynamic cable tension. The monitoring data of 14 cables in 10 days is released, with a sampling frequency of 2 Hz. Cables are numbered from left to right as SJS08 to SJS14 for the upstream side of the bridge and SJX08 to SJX14 for the downstream side of the bridge. And the dates of the released dataset are a weeklong in 2006 from 13 May to 19 May, and three separate days on 14 December 2007, 5 May 2009, and 1 November 2011. One of the released 14 cables is intended to be damaged while missing data occurred on three cables in 2011. Figure 5 illustrates the cable tension series of the released 14 cables. Sensor drifts were observed in most cables after 2007, indicating the raw data and the absolute value of cable tension are not usable directly. Sensor error was also observed in cables SJX08 and SJX13 on 1 November 2011, and in cable SJS13 on 14 December 2007, 9 May 2009, and 1 November 2011. The monitoring data is either steady or randomly jumping, so it is treated as missing data. Figure 4: The investigated cable-stayed bridge. (Red cables denote the 14 cables in the released dataset). Figure 5: _Cont_. A typical monitoring cable tension time series on a one-day scale is illustrated in Figure 6a and details on the 35 s scale are illustrated in Figure 6b. Low-frequency trend items induced by temperature and high frequency peak points items induced by vehicles can be observed. Further considering the dead load and noises, the monitored cable tension \(T_{total}\) writes as: \(T_{total}=T_{d}+T_{e}+T_{v}+T_{r}\), where \(T_{d}\), \(T_{e}\), \(T_{v}\) and \(T_{r}\) represent the effects of the dead load, the temperature, the vehicle, and the noises, respectively. In the moving concentrated force assumption, the vehicle-induced cable tension can be expressed as:  
\[ \begin{bmatrix}T_{v,1}\\ T_{v,2}\\ \vdots\\ T_{v,M}\end{bmatrix} = \begin{bmatrix}d_{11}&d_{12}&\cdots&d_{1N}\\ d_{21}&d_{22}&\cdots&d_{2N}\\ \vdots&\vdots&\vdots&\vdots\\ d_{M1}&d_{M2}&\cdots&d_{MN}\end{bmatrix}\begin{bmatrix}F_{1}\\ F_{2}\\ \vdots\\ F_{N}\end{bmatrix}  \tag{11}\]  
Figure 5: Cable tension of the investigated cables (red denotes the missing data cases). where \(T_{\mathrm{p},i}\), \(T_{\mathrm{p},j}\) is the vehicle-induced cable tension item for ith and jth cable, \(D(\cdot)=\left[d_{mn}\right]_{M\times N}\) is the discretized flexibility matrix, and \(F_{n}\) is the nth moving force loading on the discretized position. Figure 6: Typical cable tension time series and multi-sources. Considering the cable tension of ith cable under a single vehicle,  
\[T_{v,i}(t)=\eta_{i}(x_{n}(t),y_{n}(t))\cdot F_{n} \tag{12}\]  
where \((x_{n}(t),y_{n}(t))\) represents the discretized location of the vehicle on the girder and \(\eta_{i}(x_{n}(t),y_{n}(t))\) is the influence surface of the vehicle that is determined by the relative stiffness of stay cables and can be decoupled as the influence line on the longitudinal direction and the transverse direction \(\eta_{i}(x_{n}(t),y_{n}(t))=\eta_{xi}(x_{n}(t))\cdot\eta_{yi}(y_{n}(t))\). Considering multiple vehicles on the bridge, which is more frequent in realistic, the vehicle-induced item of the ith and jth cable are:  
\[