## 3 Numerical Validation Examples
### Six-Story Shear Building
#### 3.1.2 Damage Detection by the Proposed BMUA  
Changes in flexural stiffness, _EI_ (elastic modulus multiples by the second moment of inertia) and mass density are used to simulate damage cases. Table 3 shows the different damage cases in this example. The negative sign denotes the reduction of mass/stiffness. Damage detection is usually conducted with less prior information on structural parameters. Thus, unity is defined as the initial value for each mass and stiffness parameter, representing an assumption of healthy condition. When detecting damage, the simulated measured data here include six frequencies and mode shapes. Let us recall that the traditional BMUA employed the classical eigenequation in Equation (1) to update structural parameters, which inherently entails the coupling effect of mass and stiffness and may have erroneous results in both mass and stiffness identification, while the proposed BMUA \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Four Modes**} & \multicolumn{2}{c}{**Five Modes**} & \multicolumn{2}{c}{**Six Modes**} \\ \hline
**Mode** & **Actual** & **Updated** & **Updated** & **Updated** & **Updated** \\ \hline
1 & 1.0177 & 1.0018 & 1.0015 & 1.0184 \\
2 & 2.9838 & 2.9680 & 2.9645 & 2.9821 \\
3 & 4.7465 & 4.7199 & 4.7147 & 4.7467 \\
4 & 6.1858 & 6.1731 & 6.1628 & 6.1849 \\
5 & 7.2035 & 7.1720 & 7.1791 & 7.2045 \\
6 & 7.7303 & 7.7198 & 7.7258 & 7.7303 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of updated frequencies (Hz). Figure 3: Updated mode shapes using incomplete modes. uses the new eigenequations in Equations (11) and (17) in fundamentally addressing the coupling effect to accurately identify both mass and stiffness. The updated frequencies by the proposed BMUA have a highly acceptable agreement and the error is less than 2% (herein, \(\text{error}=\left|f_{updated}-f_{actual}\right|\middle/f_{actual}\), \(f_{updated}\) and \(f_{actual}\) are the updated frequency and the actual frequency, respectively), with actual values in two damage cases, as shown in Table 4 and Figure 4. In contrast, significant discrepancies between frequencies updated by the traditional BMUA and actual ones are observed: in damage case No. 1, 10.74% bias at the 6 th frequency; in damage case No. 2, 14.26% bias at the 4 th frequency. On the other hand, an excellent coincidence between the actual mode shapes and the proposed approach's identified mode shapes is exhibited in Figure 5; nevertheless, the mode shapes acquired by traditional Bayesian greatly differ from the actual mode shapes. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{} & \multicolumn{3}{c}{**Damage No. 1**} & \multicolumn{3}{c}{**Damage No. 2**} \\ \cline{2-7}  & & **Proposed** & **Traditional** & & **Proposed** & **Traditional** \\  & & **Approach** & **Approach** & & **Approach** & **Approach** \\ \hline
**Mode** & **Actual** & **Updated** & **Updated** & **Actual** & **Updated** & **Updated** \\ \hline
1 & 1.0036 & 1.0182 & 0.9288 & 0.9944 & 1.0006 & 0.9672 \\
2 & 2.9411 & 2.9379 & 2.6810 & 2.8743 & 2.8803 & 2.7870 \\
3 & 4.5735 & 4.5821 & 4.273 & 4.6106 & 4.6115 & 3.8994 \\
4 & 6.2419 & 6.2414 & 5.7089 & 5.5866 & 5.6004 & 6.3832 \\
5 & 6.9934 & 6.9941 & 6.5862 & 6.8969 & 6.8956 & 7.4558 \\
6 & 7.9430 & 7.9511 & 7.0902 & 7.4084 & 7.4125 & 7.7363 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Frequencies’ (Hz) comparison by the proposed and traditional approach. Figure 4: Updated frequencies by two approaches: (**a**) damage case No. 1; (**b**) damage case No. 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case No.** & **Mass Change** & **Stiffness Change** \\ \hline
1 & \(-\)20\% (5 th floor) & \(-\)10\% (1 st floor), \(-\)20\% (3 rd floor) \\
2 & \(-\)30\% (2 nd floor), \(-\)20\% (5 th floor) & \(-\)20\% (2 nd floor), \(-\)30\% (4 th floor), \(-\)40\% (6 th floor) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Damage cases.