## 4 Numerical Simulation
### Damage Localization of a Simply Supported Beam  
As shown in Figure 2, a simply supported beam is used for numerical simulation analysis to verify the proposed method. The beam is divided into 28 elements with the element number shown in circles. The length of the beam is 2.8 m (0.1 m for each element). Each member is 9 mm thick and 50 mm wide. The elastic modulus and the mass density of materials are, respectively, 206 GPa and 7800 kg/m\({}^{3}\). Rayleigh damping is adopted for the system with modal damping ratios \(\xi_{1}=\xi_{2}=0.01\) at the first two modal frequencies. The first six system natural frequencies are 5.51, 8.20, 23.9, 53.7, 62.9, and 82.6 Hz. The strain responses from the 9 th element to the 20 th element (the middle span of the beam, a total of 12 elements) are measured with sampling frequency of 200 Hz and duration of 30 s. White noise of 30% noise level is added to the  
Figure 1: Flowchart of damage localization approach. Figure 2: Finite element model of the simply supported beam. observation data, and the noise addition method is: \(\varepsilon^{m}(t)=\varepsilon(t)+Ep\times Noise\times std(\varepsilon(t))\), where \(\varepsilon(t)\) is the strain response obtained by simulation calculation, \(Ep\) is the noise level, _Noise_ is the standard normal distribution random number, and \(std\) is the standard deviation of \(\varepsilon(t)\). Different damage levels are simulated by reducing the stiffness of the beam element. For the case of the intact structure, Gaussian white noise with the same intensity is applied to each vertical degree of freedom (DOF) of the main beam to simulate the ambient excitation under operating conditions. The same intensity of Gaussian white noise means that their power spectrum is the same; in other words, their second-order statistical properties (including RMS) are the same. For a total of 12 elements, the ninth element is selected as a reference, and the TFs of 11 elements are obtained by Equation (4). Concerning 11 elements, the Fourier transform length here is 2048, and the first 1024 spectral points in each TF imaginary frequency curve are considered. Then, the transmissibility function sample matrix with dimension 500 \(\times\) 1024 of the intact structure is obtained by measuring 500 times with different intensities of ambient excitation. The first resonance frequency of the undamaged structure is 5.51 Hz. When MSD is solved, a small frequency band, 3.6-6.5 Hz around the first resonance frequency, that is, 31 frequency points in the first resonance frequency range, is selected for analysis. Finally, the inverse covariance matrix and the mean value of the matrix at each frequency point are obtained. In the case of a single-element damaged condition, it is assumed that (1) damage occurs in the 17 th element with a damage level of 10% subjected to El-Centro earthquake, and (2) damage occurs in the 14 th element with a damage level of 10% subjected to Chi-Chi seismic excitation. It is assumed that the El-Centro earthquake and Chi-Chi seismic excitation both act on every vertical DOF. Figure 3a,b show the seismic acceleration records. The TF of each element was calculated by Equation (6) and substituted into Equation (8). Then, the MSD damage indicator of each element is calculated. The results of the damage localization are shown in Figure 4. It is clear that the \(\text{MSD}_{\text{(test)}}\) in the 17 th element and the 14 th element significantly exceeds its threshold \(\text{MSD}_{\text{max(intact)}}\). Therefore, the 17 th element and the 14 th element can be detected as damage locations. Figure 3: (**a**) El-Centro seismic acceleration records; (**b**) Chi-Chi earthquake acceleration records. In the case of multi-element damaged condition, it is assumed that (1) the damage is introduced in the 11 th and 17 th elements with 10% stiffness reduction under El-Centro earthquake, and (2) the damage is introduced in the 13 th element with 10% stiffness reduction and the 18 th element with 15% stiffness reduction under Chi-Chi seismic excitation.