## 4 Numerical Simulation
### Damage Localization of a Cable-Stayed Bridge  
The benchmark model of the Puqian Bridge is shown in Figure 6. The length of the main girder of the bridge is 460 m (230 m + 230 m), which is divided into 236 elements. The cables are arranged in a fan shape and divided into 34 elements. The cables are connected to the main girder through motion coupling constraints. The elastic modulus of the cables is corrected by the Ernst formula according to the construction tension cable force. Table 1 summarizes the structural properties of the cable-stayed bridge model. The sampling frequency is 200 Hz, the duration is 30 s, and the strain response is observed at 1/2 of the length of the upper surface of each element. The first six system natural frequencies are 0.315, 0.993, 1.10, 1.52, 1.59, and 1.96 Hz. This paper only studies the linear behaviour of  
Figure 4: (**a**) Single-damage of the 17 th element under El-Centro earthquake; (**b**) single-damage of the 14 th element under Chi-Chi earthquake. Figure 5: (**a**) Multi-damage of the 11 th and 17 th elements under El-Centro earthquake; (**b**) multi-damage of the 13 th and 18 th elements under Chi-Chi earthquake. the structure and simulates different degrees of damage by reducing the stiffness of the beam element. For the case of intact structure, Gaussian white noise with the same intensity, that is, the same power spectrum, is applied to each vertical degree of freedom of the main beam to simulate the ambient excitation under operating conditions. Part of the strain responses (from the 50 th element to the 70 th element) is observed. The time-history of the strain response at the midpoint section of the 69 th element is shown in Figure 7a. The white noise of 30% noise level was added to the observation data as the observation response value. The power spectrum of the strain response of the intact structure under ambient excitation is estimated. The 50 th element is selected as the reference, and the TF of two measuring points is calculated by Equation (4) as an \(m\)-dimensional vector. The Fourier \begin{table}
\begin{tabular}{c c} \hline \hline
**Beam Properties** & **Value** \\ \hline Moment of inertia & 2.326 m\({}^{4}\) \\ Young’s modulus & 210 Gpa \\ Unit length mass & 17,555 kg/m \\ Damping ratio & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The main structural parameters of the main bridge. Figure 6: Schematic diagram of a two-dimensional model of a cable-stayed bridge. Figure 7: (**a**) Time history of strain response in the 69 th element under ambient excitation; (**b**) time history of strain response in the 69 th element under Chi-Chi earthquake. transform length here is 2048. The first 1024 spectral points in the imaginary frequency curve of each TF are considered. Then, a matrix with dimension \(n\times m\) is formed by using the method of numerical simulation and repeated observation for \(n\) times with different intensities of ambient excitation. After 500 simulations of ambient excitation with different intensities, a dimension of 500 \(\times\) 1024 matrix is obtained as the TF matrix of the intact structure. When solving the MSD, 11 frequency points around the first resonance frequency are selected for analysis. Finally, the mean value and inverse covariance matrix is calculated. For single damage, the damage is introduced with the level of 20% in the 69 th element under Chi-Chi earthquake. For multiple damages, the damage level of the 59 th element is 15% and that of the 64 th element is 20% under Chi-Chi earthquake. It is assumed that Chi-Chi earthquake is applied to every vertical DOF. The numerical simulation response data of the 69 th element is shown in Figure 7b, to which Gaussian white noise with a noise level of 30% is added. The results of the damage localization are shown in Figure 8. It is clear that the \(\text{MSD}_{\text{(test)}}\) of the 69 th element is obviously greater than its threshold \(\text{MSD}_{\text{max(inact)}}\).