## 7 Short-Length Energy-Saving Accelerograms for Numerical Analyses  
Since NLTHAS generally involve a high computational demand, the ground motions introduced in the numerical model should be as short as possible. On the other hand, when cutting the full recorded accelerogram, the most significant peaks should be maintained and a high enough percentage of the seismic energy of the whole event conserved. To this end, the Arias Intensity \(I_{A}\) can be exploited. The Arias intensity was introduced as an indicator of the destructive potential of an earthquake, under the assumption that the damage experienced by a structure is proportional to the energy dissipated during the seismic event. It is defined as the integral of the dissipated energy \(E(\omega)\) over the frequency domain [52]:  
\[I_{A}=\int_{0}^{\omega}E(\omega)\;d\omega \tag{24}\]  
In the time domain, \(I_{A}\) was expressed instead as the integral of the square acceleration over the complete duration \(t\) of the earthquake [45]:  
\[I_{A}=\frac{\pi}{2g}\int_{0}^{t_{d}}a^{2}(t)\;dt \tag{25}\]  
Here \(a(t)\) is the ground acceleration, g is the gravity acceleration and \(t_{d}\) is the total duration of the record. The Arias intensity defined by Equation (25) is like the formulation used by [PERSON] and [PERSON] [21] when introducing the limits of the significant duration of an earthquake. The Arias intensity can be exploited to find the most efficient and energy-conserving time-window of the earthquake. For this purpose, the SeismoSignal [53] program can be used. For any given filtered and baseline corrected ground motion, this program provides the effective duration of the earthquake corresponding to values of \(I_{A}\) from 5% to 95%. In the following, an example is given to illustrate how SeismoSignal can lead to obtaining a suitable time-window of a given earthquake. Let us refer to the two components of one of the IBC-spectrum-consistent earthquakes displayed in Figure 7, namely IBC-C-X-E5 and IBC-C-Y-E5. Once the accelerograms (see Figure 10a) are uploaded in SeismoSignal and the option \"effective duration\" is set, the values given in Table 10 can be extracted. The diagrams in Figure 10b can also be extracted, where the time-windows relevant to the 5-95% range of \(I_{A}\) are highlighted in green. It should be noted that the starting and the final values of the time-windows are different for the two components, as can be inferred from Figure 10c. Since same-duration components of the earthquake must be given for the analyses, a single time-window should be set. An effective and energy-saving time-window could be defined by choosing the lowest time value and the highest time value of the two time-windows found through the Arias intensity procedure, see Figure 10d. Eventually the full-length accelerograms of Figure 10a are reduced to those shown in Figure 10e which are saving 95% of the earthquake energy, while being shortened to about \(\frac{1}{4}\) of the full length. It is noted, in fact, that saving almost all the energy of the original ground motion is of paramount importance since the effects of ground motions on structures are strongly related to the energy input [54]. To validate the procedure, the planar four-story building displayed in Figure 11 was adopted as a simple archetype. The structural elements are made of reinforced concrete, with square columns measuring 50 cm on each side, and beams with a height of 50 cm and a base of 30 cm. Since the purpose of this validation is not to assess the design of the elements, symmetrical longitudinal reinforcement has been assumed for the columns, consisting of 12 bars of 20 mm and transverse bars of 10 mm spaced evenly every 10 cm. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Earthquake Component** & **Duration [s]** & **Cumulative \(I_{A}\) [m/s]** & **Percentage of \(I_{A}\) [\%]** \\ \hline  & 6.19 & 0.3 & 5 \\ IBC-C-X-E5 & 11.58 & 6.2 & 95 \\  & 6.81 & 0.2 & 5 \\ IBC-C-Y-E5 & 7.32 & 4.1 & 95 \\ \hline \hline \end{tabular}
\end{table}