## 5 RC Frame Model Used for Model Uncertainty Quantification
### Dynamic Non-Linear Time History Analysis (NTHA)  
To determine the dynamic capacity curve, an IDA is carried out by successively increasing the vertical imposed loads on the structure by \(P_{i+1}=P_{i}+\Delta P\), where \(\Delta P\) is the load increment. A NTHA is applied for each load level, fully taking into account the dynamic effects. This gives the most accurate results, but one NTHA can only obtain the dynamic response for one imposed load case for a structural system and thus a series of NTHA (i.e., an IDA) is needed to determine the dynamic capacity curve. As the dynamic structural response becomes highly non-linear when approaching the ultimate load-bearing capacity, the load increment \(\Delta P\) of the IDA is gradually reduced. To accurately determine the resistance, a final load increment resolution of 0.05 kN/m is applied near the load for which failure occurs. Note that shear failure cannot be reproduced with the current model. For every load level, one NTHA is executed, in which the uniform loads on all the beams are first applied, followed by the instantaneous removal of column A, B, or C (a removal time duration of 0.001 s). The explicit Kolay-Ricles-alpha algorithm in OpenSees is employed to execute the dynamic column removal analysis, for which the time step is set as 0.001 s [46]. The dynamic response of the first 4 s for each NTHA is recorded at the top of column A, B, or C (at the control point). Two different values for the damping ratio are investigated in the dynamic analyses (DR = 0% and DR = 5%, where 5% is a representative value for RC structures and Rayleigh damping is adopted) [16; 67]. As the EBM cannot take into account any dynamic effects, DR = 0% in IDA is therefore investigated for a comparison purpose. Actually, the following results show that almost identical results are obtained between EBM and IDA if DR = 0%. The time-history displacement responses at the control points from the IDA when DR = 5% are presented in Figure 9a, Figure 10a, and Figure 11a for Case A, Case B, and Case C, respectively. The system oscillates around the equilibrium position and the oscillation decays due to the damping effect after the sudden column removal. Ultimate load-bearing capacities (and corresponding maximum displacements) of 35.5 kN/m (394.4 mm), 38.6 kN/m (356.9 mm), and 38.3 kN/m (341.4 mm) are obtained for Case A (Figure 9a), Case B (Figure 10a), and Case C (Figure 11a), respectively. A summary of the results is presented in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & **Parameter** & **Units** & **Mean Value** \\ \hline \multirow{4}{*}{Concrete} & Compressive strength\(f_{\rm c}\) & MPa & 28 \\  & Compressive peak strain \(\varepsilon_{\rm c1}\) & \% & 0.21 \\  & Tensile strength\(f_{\rm c1}\) & MPa & 2.2 \\  & [PERSON]’s modulus \(E_{\rm ci}\) & GPa & 30.3 \\ \hline \multirow{4}{*}{Steel} & Yield stress \(f_{\rm y}\) & MPa & 560 \\  & Tensile strength \(f_{\rm u}\) & MPa & 655 \\ \cline{1-1}  & Ultimate strain \(\varepsilon_{\rm u}\) & \% & 12 \\ \cline{1-1}  & [PERSON]’s modulus \(E_{\rm s}\) & GPa & 205 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of the materials. It is found that the failure occurs due to concrete crushing at the beam ends. However, different failure locations and sequences are observed in the three cases. Regarding Case A, the first failure (at the ultimate load-bearing capacity of 35.5 kN/m) occurs at the beam end in the first floor, followed immediately by failures at the beam ends in the fourth, third, and second floors. For Case B, the first failure (at the ultimate load-bearing capacity of 38.6 kN/m) is observed at the beam end in the first floor, followed immediately by failures at the beam ends in the fifth floor and then the other floors. In terms of Case C, the first (local) failure occurs at the beam end in the first floor with an imposed load (displacement) of 37.9 kN/m (279.2 mm) but the other floors are intact.