## 3 Procedure to Generate Probable Multi-Hazard Scenarios in Building Lifetime  
The likely recurrence of earthquake and wind events during the building lifetime is crucial to be addressed, as it enables the decision-makers to evaluate the building performance in any specific service period. Thus, a practical procedure to estimate the likelihood of the occurrence of multi-hazard scenarios in the structural design life is developed and presented by [PERSON] et al. [35]. The procedure is based on the Poisson process, which uses the anticipated return period and intensity to generate the earthquake and wind events. From the Gutenberg-Richter equation, the annual occurrence rate, \(N(I)\), of earthquake loads with an intensity more than \(I\) is given as,  
\[\ln N(I)=a-bI. \tag{1}\]  
The average recurrence period, \(T(I)\), satisfies the following relationship,  
\[N(I)=\frac{1}{T(I)}. \tag{2}\]  
If we consider two earthquake events having the intensity of \(I_{100}\) and \(I_{500}\), with the correspondent return periods of 100 and 500 years, respectively, the constants, \(a\) and \(b\), of the Gutenberg-Richter equation can be obtained as given in Equations (4) and (5) and Table 4.  
\[\left\{ \begin{array}{l}\ln(0.01)=a-bI_{100}\\ \ln(0.002)=a-bI_{500}\end{array} \right\} \tag{3}\]  
Figure 8: Ductility demand and energy dissipation rate of BRBs along the height of a high-rise steel building under earthquake and wind loads. \[b =\frac{\ln(0.01)-\ln(0.002)}{I_{500}-I_{100}} \tag{4}\] \[a =\ln(0.01)+bI_{100} \tag{5}\]  
Now, if \(N(I_{0})\) is an earthquake event with intensity more than the \(I_{0}\), the cumulative distribution function (CDF) of the intensity \(I\) is given as  
\[F(i)=P(I\leq i)=1-\frac{N(i)}{N_{0}}, \tag{6}\]  
where \(N_{0}=N(I_{0})\). Once the probability of earthquake intensity is determined, the probability of earthquake occurrence, \(P(t)\), assuming that the earthquake events occur individually with an average occurrence rate \(N(I)\), can be estimated. The probability of \(n\) earthquake events of intensity more than \(I\) in the \(t\)-years is expressed as the Poisson process as  
\[P_{n}(t)=\{N(I)t\}^{n}\ \ \frac{e^{-N(I)t}}{n_{t}}. \tag{7}\]  
In Equation (7), setting \(n=0\), the probability of no earthquake events of intensity more than \(I\) in the \(t\)-years can be determined as  
\[P_{0}(t)=e^{-N(I)t}. \tag{8}\]  
Similarly, the probability of occurrence of more than one earthquake event exceeding intensity \(I\) in the \(t\)-years could be given as  
\[P(t)=1-P_{0}(t)=1-e^{-N(I)t}. \tag{9}\]  
The probabilistic model of the occurrence of wind events is created in the same way. In order to generate the events in the expected lifetime, the proposed procedure initially consists of estimation of the probable earthquake and wind intensity, and subsequently, for the estimated intensity, calculating the probable occurrence with the generation of earthquake and wind events. In the first step, the sample of wind or earthquake intensity, \(I_{i}\) (i = 1, 2,.., \(L\)), is generated from the CDF, Equation (6), using the uniform random variable \(F_{i}\), where \(F_{i}\in[0,1]\). In conjunction, the sample of the returned period of intensity, \(i_{T}j\) (\(j=1\), 2,.., \(M\)), is generated from the CDF, Equation (8), using the uniform random variable \(P_{i}\), where \(P_{i}\in[0,1]\). Lastly, the \(L\) events with different intensity, \(I_{i}\) (\(i=1\), 2,.., \(L\)), is arranged corresponding to the \(M\) return period, \(i_{T}j\) (\(j=1\), 2,.., \(M\)). For the multi-hazard scenarios, the earthquake and wind events are assumed to occur independently, where the final timeline is added together, as further illustrated in the proposed procedure's flowchart; see Figure 9.