## 4 Validation and Case Study
### Validation Using a 3D Shaking Table Test of a Three-Story Reinforced Concrete Frame
#### 4.1.2 Comparison of Simulation Results  
Based on the information above, an FE model with fiber beam-column and multi-layered shell elements was established in MSC_Marc. Details of these element models have been reported in the published work [35]. The seismic responses of the structure under Load Cases 1-3 are used to validate the rationality of the FE model. Figure 9 shows the comparison of the maximum horizontal displacement in the \(X\)-direction between the test results and the FE simulation. The results indicate that the FE simulation agrees well with the test results.  
In Load Case 5, a convergence problem occurred in the FE simulation (\(t=4.16\) s); therefore, the distribution of the ruins could not be obtained. The collapse process predicted by the FE simulation is shown in Figure 10. The colored contours in Figure 10 represent the longitudinal reinforcement yielding in the elements. The numerical simulation reveals that failures of the frame are initiated at both ends of the columns where the bending moment is large. Subsequent failures take place on the ground story, where the columns carry the largest lateral force. Large displacements occurred on the ground story, which leads to the collapse of the entire structure. In Figure 10c, the deformed shape of the structure at 4.16 s is shown.  
\begin{table}
\begin{tabular}{c c c} \hline
**Floor No.** & \(\mathbf{E_{\text{c}}}\) **(GPa)** & \(\mathbf{f_{\text{c}}}\) **(MPa)** \\ \hline
1 & 25 & 22.5 \\
2 & 27 & 23.9 \\
3 & 23 & 15.6 \\ \hline \end{tabular}
\end{table}
Table 1: Concrete information.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Load Case** & \multicolumn{3}{c}{**Peak Ground Acceleration (PGA) (Unit: g)**} \\ \cline{2-4}  & \(X\)**-Direction** & \(Y\)**-Direction** & \(Z\)**-Direction** \\ \hline
1 & 0.10 & 0.08 & 0.06 \\
2 & 0.36 & 0.32 & 0.28 \\
3 & 0.84 & 0.55 & 0.54 \\
5 & 0.93 & 1.19 & 0.57 \\ \hline \end{tabular}
\end{table}
Table 3: Load cases.  
Figure 8: Pseudo-acceleration spectra of Load Case 5.  
Figure 10: Collapse process of the reinforced concrete frame subjected to the El-Centro ground motion record: (**a**) \(t=0.00\) s; (**b**) \(t=3.88\) s; (**c**) \(t=4.16\) s.  
Figure 9: Maximum horizontal displacement in the X-direction: FE method versus test.  
From the FE results, the initial moment of collapse is 3.88 s, using the collapse criterion in Section 3.2. A \(\frac{1}{4}\)-scale model is adopted; thus, the vertical collapse displacement of the collapse criterion is 0.25 m.  
Subsequently, collapse simulation was performed using the proposed hybrid method. Figure 11b shows the distribution of ruins, which shows excellent agreement with the test results.  
When BCB is used for collapse simulation, the structure collapses vertically, which agrees poorly with the test results, due to the low accuracy of the BCB simulation of the small-deformation stage. As shown in Figure 11c, the final positions of the three slabs of the three-story reinforced concrete frame almost entirely overlap.  
Compared with the BCB results (Figure 11c), the distribution of the ruins obtained by the proposed hybrid method (Figure 11b) shows better agreement with the test results (Figure 11a).  
Figure 11: Distribution of ruins: (**a**) test; [20] (Reproduced with permission from [PERSON], from _Study on spatial collapse responses of reinforced concrete frame structures under earthquake_; published by Tongji University, 2006). (**b**) the proposed FE and BCB hybrid method; (**c**) BCB.  
The following observations are made from the comparison and analysis of the simulation methods:  
* For FE models, a convergence problem occurs during the collapse simulation; therefore, the distribution of ruins cannot be easily obtained.
* The BCB simulation method is not sufficiently accurate for the small-deformation stage. Therefore, the collapse mode and the distribution of ruins differ substantially from the test results.
* The proposed hybrid method achieves the most satisfactory simulation of the structural collapse mode and the distribution of ruins.