## 2 Average Annual Collapse Probability and Collapse Safety Margin Ratio
### Average Annual Collapse Probability of Structures  
The annual collapse frequency of exceedance \(\lambda_{collapse}\) of structures can be obtained by coupling structural collapse vulnerability \(P(C|IM)\) with seismic risk \(\lambda(IM)\), as shown in Equation (1), which can be deemed equal to the average annual collapse probability when it is smaller than \(10^{-2}\)[18].  
\[\lambda_{collapse}=\int P(C|IM)\left|\frac{d\lambda(IM)}{d(IM)}\right|d(IM) \tag{1}\]where \(IM\) is the earthquake intensity parameter. The earthquake intensity at which structures collapse is supposed to conform to the lognormal distribution [26; 27], and the spectral acceleration at fundamental period \(S_{a}\)(T\({}_{1}\)) is the earthquake intensity parameter \(IM\). Then, the structural collapse probability \(P(C|IM)\) becomes  
\[P(C|IM)=\Phi\bigg{(}\frac{\ln(S_{a})-\ln\hat{S}_{a,c}}{\sigma_{Sa,c}}\bigg{)}= \int\frac{1}{\sqrt{2\pi}\cdot\sigma_{Sa,c}\cdot S_{a}}\cdot\exp\Bigg{(}-\frac{ \left(\ln(S_{a})-\ln\hat{S}_{a,c}\right)^{2}}{2\sigma_{Sa,c}^{2}}\Bigg{)}d(S_{a }), \tag{2}\]  
where \(\ln\hat{S}_{a,c}\) is the logarithmic mean. \(\hat{S}_{a,c}\) is the earthquake intensity corresponding to 50% collapse probability. Structural collapse probabilities are acquired through IDA. \(\sigma_{Sa,c}\) is the logarithmic standard deviation and represents the uncertainty in structural collapse [27]. All the so-called total collapse uncertainties mentioned hereafter refer to \(\sigma_{Sa,c}\).  
The occurrence of earthquakes is supposed to be subjected to Poisson distribution, and the average annual frequency of exceedance of the earthquake intensity \(S_{a}\) is \(\lambda(S_{a})=k_{0}\cdot\left(S_{a}\right)^{-k}\), where \(k_{0}\) and \(k\) are two empirical coefficients shown in Equations (3) and (4) and can be determined by referring to Literature [28]; when incorporating Equation (2), Equation (1) can be simplified as Equation (5).  
\[k=\frac{\ln(\lambda_{DBE}/\lambda_{MCE})}{\ln(IM_{MCE}/IM_{DBE})}, \tag{3}\]  
\[k_{0}=\lambda_{DBE}(IM_{DBE})^{k}, \tag{4}\]  
\[\lambda_{collapse}=k_{0}\cdot\left(\hat{S}_{a,c}\right)^{-k}\cdot\exp\bigg{(} \frac{2}{2}k^{2}\cdot\sigma_{Sa,c}^{2}\bigg{)}, \tag{5}\]  
where \(IM_{DBE}\) and \(IM_{MCE}\) are the design basis earthquake intensity (exceedance probability is 10% in 50 years) and MCE intensity (exceedance probability is 2-3% in 50 years), respectively. \(\lambda_{DBE}\) and \(\lambda_{MCE}\) are the annual frequency of exceedance of the design basis earthquake and MCE intensities, respectively. The seismic hazard curve is shown in Figure 1a. When \(k\) = 2.31, \(k_{0}=1.11\times 10^{-5}\), which is calculated in accordance with an assumed RC structure whose design basis earthquake and MCE intensities are \(S_{a,\text{DBE}}\) = 0.1029 g and \(S_{a,\text{MCE}}\) = 0.21 g, respectively; \(k\) is the slope of the seismic hazard curve in the logarithm coordinate shown in Figure 1b.  
Equation (5) presents that the average annual collapse probability equals to a seismic risk term of the logarithmic mean of earthquake intensity multiplied by an exponential term. The exponential term implies that the average annual collapse probability increases with the increasing collapse uncertainty and slope of seismic hazard curve in the logarithmic coordinate.  
Figure 1: Seismic hazard curves: (**a**) Seismic hazard curve in the linear coordinate; (**b**) Seismic hazard curve in the log–log coordinate.