## 3 Seismic Action
### Earthquake Scenario
#### 3.2.2 Adjusted Response Spectrum  
When assessing a high number of individual buildings, as was the case of the school buildings included in the database of the PERSISTAH software, with hundreds of buildings that were meant to be studied for seismic retrofitting purposes, it is important to use a fast and accurate enough method. For the earthquake scenario option, this is much more important. Because the buildings' seismic safety evaluation process was meant to be closely related to the EC8 approaches, namely by using NSA methods, it was desirable to adopt an EC8-like response spectrum. Moreover, that response spectrum should match, as much as possible, the \(N_{T}\) spectral coordinates obtained throughout the GMPEs. Therefore, it was necessary to develop an algorithm to satisfy two major objectives, simultaneously, which is not an easy task: It should be as fast and accurate as possible. To adjust an EC8-like response spectrum to the results of the GMPEs, the following mathematical optimization problem is proposed:  
\[\text{Minimize}\ F(\alpha_{a},T_{B},T_{C},T_{D})=\sum\
olimits_{i=1}^{N_{T}} \left(S_{e,GMPE(T_{i})}-S_{e,EC8(T_{i},a_{a},T_{B},T_{C},T_{D})}\right)^{2}, \tag{1}\]  
Subject to  
\[0\leq T_{B}\leq T_{C}\leq T_{D}\leq 4\ s, \tag{2}\]  
Figure 4: Types of possible sources and distances to the earthquake: (**a**) Point source; (**b**) line source; (**c**) fault plane (rectangular). \[\alpha_{a}\geq 0, \tag{3}\]  
where \(T_{B}\), \(T_{C}\), \(T_{D}\) are the periods of the response spectrum that are established in the EC8, and \(\alpha_{a}\) is a parameter that reflects the maximum spectral amplification (which is equal to 2.5 in the EC8, but in this work it was left to the optimization process to compute that value). The EC8 spectrum is a parametric function, given by  
\[S_{e,\text{EC8}(T,\alpha_{a},T_{B},T_{C},T_{D})}=\left\{ \begin{array}{ll}a_{ g}\cdot S\left[1+\frac{T}{T_{B}}\cdot(\eta\cdot\alpha_{a}-1)\right]&,\,\,0\leq T \leq T_{B}\\ a_{g}\cdot S\cdot\eta\cdot\alpha_{a}&,\,\,T_{B}\leq T\leq T_{C}\\ a_{g}\cdot S\cdot\eta\cdot\alpha_{a}\frac{T_{C}\cdot T_{D}}{T^{2}}&,\,\,T_{D} \leq T\leq 4\,s\end{array} \right. \tag{4}\]  
with \(S\) being the soil factor and \(\eta\) a correction factor, which is a function of damping (\(\xi\) in percentage)  
\[\eta=\sqrt{\frac{10}{5+\xi}}\geq 0.55. \tag{5}\]  
Observing Equations (1)-(3), it is possible to conclude that this optimization problem has a nonlinear objective function, but the problem restrictions are linear, so it is important to keep this in mind when selecting an optimization algorithm, which should be accurate and fast enough as mentioned above. In Figure 5, an example of an EC8-like response spectrum adjusted to the results of a GMPE is presented. In the implemented approach, the optimization problem is solved by using a variation of the complex method [47; 48] (Figure 6), which seems to work properly to obtain the solution, just by looking to Figure 5. However, some care was taken in the parameter definition and in the determination of the initial admissible solutions to obtain better results. The values of the parameters presented in  
Figure 5: Input/output ground-motion prediction equations (GMPEs) user interface with an ECS-like adjusted response spectrum. Figure 6 that were adopted as default are \(k=16\), \(\alpha=1.3\), \(n_{lim}=7\) and \(k_{lim}=18\), which were obtained by a trial and error process in order to achieve a compromise between speed and accuracy. For the initial admissible solutions, the vectors are randomly selected so that \(T_{B}\) is always lower than the period corresponding to the maximum spectral acceleration, and \(T_{C}\) is always higher. Many pairs of magnitude and distance were used to test the implemented optimization procedure. Results seem to indicate that good fitting was possible to obtain (see the example of Figure 5).