## 7 Numerical Results of NSDI
### 120-Bar Dome Truss  
The second example is a 120-bar dome truss, as illustrated in Figure 5 [48].  
Figure 5: Diagram of a 120-bar dome truss: (**a**) plan view, (**b**) section view.  
The dome's diameter and height are 31.78 m and 7 m, respectively. A seamless steel pipe with a modulus of elasticity of 30,450 ksi (210,000 MPa) and material density of 0.288 \(\frac{lb}{in^{3}}\) (7971.810 \(\frac{kg}{m^{3}}\)) is used. The pipes have an external diameter of 0.2 m and a thickness of 0.006 m. For FE modeling and analysis, OpenSees software [42] was used. As described in Table 7, two damage scenarios are considered. Figures 6 and 7a and Tables 8 and 9 show the performance of the proposed method. It is worth noting that the noise level is set as 2% and 5% in the frequencies and mode shapes, respectively.  
The results show that although the measured modal data were noise corrupted, the proposed approach provides a reliable prediction of damage occurrence. Furthermore, ESMA can effectively find the correct locations and severity of the damage. For a larger noise level, that is \(\xi_{\lambda}=10\%\), the resultant PoDE values of damage scenario 2 are shown in Figure 7b and Table 10.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Case 1**} & \multicolumn{3}{c}{**Case 2**} \\ \hline
**Element Number** & **SRF** & **Element Number** & **SRF** \\ \hline
12 & 0.30 & 4 & 0.35 \\
38 & 0.20 & 30 & 0.20 \\
53 & 0.25 & 51 & 0.35 \\
79 & 0.2 & 58 & 0.25 \\ - & - & 89 & 0.2 \\ - & - & 105 & 0.40 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Damage scenarios for the 120-bar dome truss.  
Figure 6: Probability of damage existence for damage case 1 of the dome truss.  
Figure 7: Probability of damage existence for damage case 2 of the dome truss. (**a**) Probability of damage existence with noise level 5%. (**b**) Probability of damage existence with noise level of 10%.  
The numerical results show that the proposed method performs well in detecting the location and severity of the damage scenarios even with a large noise level. The PoDE values of the damaged elements are higher in NSDI approaches, whereas the PoDE values of the undamaged elements are lower.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Element Number** & **PoDE** & **PDE** \\ \hline
2 & 0\% & 3\% \\
3 & 7\% & 2\% \\
4 & 88\% & 83\% \\
5 & 0\% & 1\% \\
30 & 92\% & 78\% \\
31 & 2\% & 10\% \\
32 & 4\% & 9\% \\
33 & 10\% & 0\% \\
50 & 7\% & 10\% \\
51 & 87\% & 80\% \\
52 & 14\% & 12\% \\
56 & 0\% & 1\% \\ \hline \hline \end{tabular}
\end{table}
Table 10: PoDE for damage case 2 of the dome truss with \(\xi_{\lambda}=10\%\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Element Number** & **PoDE** & **PDE** \\ \hline
10 & 2\% & 9\% \\
11 & 0\% & 4\% \\
12 & 98\% & 94\% \\
13 & 0\% & 8\% \\
37 & 0\% & 4\% \\
38 & 95\% & 96\% \\
39 & 10\% & 14\% \\
51 & 0\% & 3\% \\
53 & 98\% & 95\% \\
78 & 3\% & 8\% \\
79 & 95\% & 85\% \\
80 & 0\% & 10\% \\ \hline \hline \end{tabular}
\end{table}
Table 8: PoDE for damage case 1 of the dome truss.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Element Number** & **PoDE** & **PDE** \\ \hline
10 & 2\% & 9\% \\
11 & 0\% & 4\% \\
12 & 98\% & 94\% \\
13 & 0\% & 8\% \\
37 & 0\% & 4\% \\
38 & 95\% & 96\% \\
39 & 10\% & 14\% \\
51 & 0\% & 3\% \\
53 & 98\% & 95\% \\
78 & 3\% & 8\%