## 3 Numerical Validation Examples
### Three-Dimensional Three-Story Shear Building  
A three-dimensional three-story shear building is utilized to evaluate the proposed BMUA under more complex conditions, including damage detection and comparative investigation between the traditional and the proposed BMUA in Section 3.2.1. Finally, probabilistic damage estimation is provided to predict structural failure in Section 3.2.2. The diagram of the structural model is shown in Figure 8. In this example, the Young's modulus and mass density are assumed to be 69 GPa and 2700 kg/m\({}^{3}\), respectively. Each floor has the length and width of both 4.3 m with 0.2 m thickness, giving the mass weight of M = 10\({}^{4}\) kg. Hence, a total of three mass parameters are to be updated. Four stiffness values at each floor are assumed, giving twelve-be-updated stiffness parameters as \(\theta_{4(j-1)+1}=K_{j,+x}\), \(\theta_{4(j-1)+2}=K_{j,+y}\), \(\theta_{4(j-1)+3}=K_{j,-x}\), \(\theta_{4(j-1)+4}=K_{j,-y}\), \(j=1,2,3\), where \(j\) denotes the story number and \(+x\), \(-x\), \(+y\) and \(-y\) are the directions of the structural outer face. All columns have the same material and geometric properties with the length and moment of inertial of 5 m, 0.006 m\({}^{4}\) (\(y\)-direction) and 0.0075 m\({}^{4}\) (\(x\)-direction), respectively. The nominal magnitudes of inter-story stiffness are estimated to be \(K_{j,+y}=K_{j,-y}=40,000\) kN/m and \(K_{j,+x}=K_{j,-x}=50,000\) kN/m. Figure 7: Probabilistic curves. Damage case No. 1: (**a**) mass; (**b**) stiffness. No. 2: (**c**) mass; (**d**) stiffness. An earthquake-damaged structure is repaired using a typical retrofitting technique. Let us suppose that the frame model is subject to unknown seismic activities, yielding severe structural damage, such as cracks and bearing deterioration, further impairing stiffness and ductility. Therefore, it is essential to repair the structure and strengthen its resistance capability to avoid any collapse. Herein, the Buckling-restrained trace (BRB) (see Figure 1c) is a useful seismic retrofit method and can provide additional stiffness and energy dissipation capacity [46]. In this work, twelve BRBs marked as red color are welded on the structure in four directions, as shown in Figure 8b. Assuming each BRB installed in the \(x\)- or \(y\)-direction has the same sectional and material properties (length of 6.59 m, moment of inertial of 0.0006 m\({}^{4}\) (\(x\)-direction) and 0.0005 m\({}^{4}\) (\(y\)-direction)), to provide the same stiffness modification, \(\Delta k_{j,+x}=\Delta k_{j,-x}=1750\) kN/m, \(\Delta k_{j,+y}=\Delta k_{j,-y}=1400\) kN/m on each floor, the weight of each BRB is ignored here. In this example, the measured data contain the first six frequencies and mode shapes in the original and modified system with stiffness modification. Gaussian white noise with zero-mean and 1% COV of modal parameters is considered on the measured data for both the proposed BMUA and the traditional BMUA. Table 11, the mass parameter and the stiffness parameter, \(\beta_{3}\) and \(\theta_{3,-y}\), are updated as 0.0078 and 1.3603 (target: 0.7000 and 1.0000). As seen in Figure 10, the mass/stiffness parameters identified by the proposed approach highly agree with the target values for the considered damage cases. Only an error of less than 3% can be observed. However, mass/stiffness parameters are updated with prominent discrepancies by the traditional Bayesian approach; even some updated results are completely unacceptable, such as too-small values in the updated mass parameters. Similar to the first example, the traditional Bayesian approach fails to update both mass and stiffness parameters using the classical eigenequation in Equation (1), resulting in poor or false damage detection. It can be attributed to the coupling effect of mass and stiffness existing in traditional Bayesian that governs the accuracy of the updated results when simultaneously updating mass and stiffness.