## 3 Acceptable CMR and Collapse Uncertainty
### Uncertainty in Collapse Analysis  
Performance-based earthquake engineering should implement a probabilistic prediction of the seismic response of structures with consideration to multiple sources of uncertainty [29]. The inclusion of uncertainty plays an equally important role in the collapse performance assessment of retrofitted structures.  
Structural collapse-resistant capability is influenced not only by the uncertainty in the input ground motion, i.e., record-to-record variability, but also by factors such as model uncertainties \(\sigma_{MDL}\), design requirement uncertainties \(\sigma_{DR}\), and test data uncertainties \(\sigma_{TD}\). The total collapse uncertainty \(\sigma_{S_{a,c}}\) can be calculated from these uncertainties by [9]  
\[\sigma_{S_{a,c}}=\sqrt{\sigma_{RTR}^{2}+\sigma_{DR}^{2}+\sigma_{MDL}^{2}+\sigma _{TD}^{2}}. \tag{10}\]  
With the given structural collapse probability, an acceptable CMR is a random variable that varies with the total collapse uncertainty. The magnitude of uncertainty influences the calculation of collapse safety margin by changing the shape of collapse vulnerability curves.  
Figure 4 shows a collapse vulnerability curve considering total collapse uncertainty \(\sigma_{S_{a,c}}\) (dashed line). The figure also shows a collapse vulnerability curve considering only record-to-record variability \(\sigma_{RTR}\) (solid line). The dashed curve reflects a \(\sigma_{S_{a,c}}=0.8\), and the solid curve reflects a \(\sigma_{RTR}=0.55\); these values are calculated using the IDA method with 50 ground motions. The collapse probability of the MCE intensity of the former is increased. For normally designed ductile structures, MCE intensity is less than the median collapse intensity. Therefore, an increase in collapse uncertainty elevates the MCE collapse probability.  
Figure 3: Design acceleration response spectrum.  
Among all the four above-mentioned uncertainties, record-to-record variability \(\sigma_{RTR}\) is relatively easy to determine because it can be directly obtained through IDA [30]. Other uncertainties are determined mostly with experience. Thus, reasonably determining the collapse uncertainty coefficients is critical. However, relative statistical data and research in China are still desired comparatively at present, and the existing ones are unsystematic; limited research has focused on model uncertainty [31; 32]. [PERSON] and [PERSON] [31] conducted a collapse vulnerability analysis on an RC frame structure considering record-to-record variability and structural model uncertainty. The presence of structural uncertainty increased the logarithmic standard deviation by 70%. Through the parametric analyses of 10 models with different loads, concrete grades, and viscous damping ratios, [PERSON] et al. [32] found that the uncertainty in model parameters equals to approximately 50%-65% of record-to-record variability. Although the adopted structural model parameters in the two studies are the same, the calculation methods applied differ and lead to differences in results. Model parameters also have different influences on structural collapse sensitivity; in other words, no universal value exists for model uncertainty.  
The determination of uncertainties other than record-to-record variability \(\sigma_{RTR}\) should refer to the suggestion of FEMA P695 due to the absence of clear categories for collapse uncertainties in Chinese codes. Despite the differences between the Chinese and American seismic codes, the macroscopic quality requirements on the uncertainties in FEMA P695 are similar to the considerations in the structural design and analysis in Chinese codes. Hence, referring to the macroscopic quality requirement in FEMA P695 is appropriate to control the levels of uncertainty and then quantify the uncertainties when designing according to Chinese specifications.