# Structural Damage Localization by the Principal Eigenvector of Modal Flexibility Change
## 3 Numerical Example  
The continuous beam in Figure 4 is used to demonstrate the defect localization ability of the presented PE algorithm. Note that this example is a statically indeterminate structure. For the structure, the elastic modulus, density, moment of inertia, and cross-sectional area are 200 GPa, 7.8 \(\times\) 10\({}^{3}\) Kg/m\({}^{3}\), \(1.0416\times 10^{-6}\) m\({}^{4}\), 0.0025 m\({}^{2}\), respectively. In the FEM, the structure is divided into 36 elements, which gives 33 translational DOFs and 37 rotational DOFs. The length of each segment in the beam FEM is 0.1 m. In the following discussion, only the first four modal parameters with the translational degrees of freedom are used in order to simulate the incomplete measurements. Defect in the beam was simulated by reducing the elastic modulus of some structural elements. Five defect scenarios are assumed in the example. Scenario 1: the defect occurred in element 7 with 20% stiffness reduction. Scenario 2: the defect occurred in element 13 with 20% stiffness reduction. Scenario 3: the defects occurred in elements 5 and 17 both with 20% stiffness reduction. Scenario 4: the defects occurred in elements 5 and 31 both with 20% stiffness reduction. Scenario 5: the defects occurred in elements 7, 19 and 33 all with 20% stiffness reduction. Detection results of each damage scenario obtained  
Figure 4: A beam structure with three spans. by the presented PE algorithm and the well-known ULS method are both given to compare the localization performance. Using the exact data without noise, the principal eigenvector (PE) of modal flexibility change and the ULS variation for damage scenario 1 are plotted in Figure 5a,b, respectively. Accordingly, the PE curvature and the ULS variation curvature for the case of damage scenario 1 are plotted in Figure 6a,b, respectively. Figure 5: (a) Damage index (principal eigenvector (PE)) map using the first four exact modes when the defect occurred in element 7; (b) Damage index (uniform load surface (ULS) variation) map using the first four exact modes when the defect occurred in element 7 (\(\times 10^{-4}\)). Figure 6: (a) Damage index (PE curvature) map using the first four exact modes when the defect occurred in element 7; (b) Damage index (ULS variation curvature) map using the first four exact modes when the defect occurred in element 7 (\(\times 10^{-3}\)). One can see that the transnational DOFs numbered 7 and 8 are the turning points in Figure 5 and the sharp peaks in Figure 6, respectively. In the beam model, the seventh and the eighth transnational DOFs are exactly corresponding to the seventh element. This means that the seventh segment could be successfully detected to be the defect area by inspecting the turning points in Figure 5, or by inspecting the sharp peaks in Figure 6. Note that the human inspection and intervention is required in the above damage localization process. Damage scenario 2 is used to verify the validity of the two algorithms when the defect occurred near the supported boundary. Figures 7 and 8 showed the detection results of this scenario. One can conclude that the damaged element 13 can be detected by inspecting the turning points in Figure 7, or by inspecting the sharp peaks in Figure 8.  
Damage cases 3-5 are used to compare the PE algorithm with the ULS  
Figure 8: **(a)** Damage index (PE curvature) map using the first four exact modes when the defect occurred in element 13; **(b)** Damage index (ULS variation curvature) map using the first four exact modes when the defect occurred in element 13 (\(\times 10^{-3}\)). Figure 9: **(a)** Damage index (PE) map using the first four exact modes when the defects occurred in elements 5 and 17; **(b)** Damage index (ULS variation) map using the first four exact modes when the defects occurred in elements 5 and 17 (\(\times 10^{-4}\)). Figures 11-14 presented the damage localization results for damage case 4 using the exact modal data and the data with 3% noise, respectively.