## 4 Trends of Damage as a Function of Seismic Intensity  
In this section, it is investigated whether the level of a seismic intensity measure (IM) can be associated with the level of damage (in terms of rotation) of the steel beams for the considered hypothetical buildings. The pseudo-spectral acceleration, Sa, is selected to inspect possible correlations with the beams rotations, because this IM is readily available in engineering practice and easier to communicate to decision makers. First, the horizontal accelerations for the three vibration modes leading to the highest mass participation factors \(\Gamma_{h}\) are obtained. This information is listed in Table 4 for the 6-story building. It is pointed out that not necessarily the three first modes lead to the highest participation factors; whereas this is usually the case for the horizontal component, it is not for the vertical component. Then the square root of the sum of the squares (SRSS) of the horizontal accelerations is computed (denoted as \(Sa\)\(H_{SRSS}\)) with the following expression:  
\[Sa\ H_{SRSS}=\sqrt{\sum_{j=1}^{n}\frac{1}{n}\Big{[}\Gamma_{hj}\times Sa\big{(} Th_{j}\big{)}\Big{]}^{2}} \tag{1}\]  
where n is the number of modes selected (i.e., 3 in this study), \(\Gamma_{hj}\) is the mass horizontal modal participation factor for the \(j\) mode and \(Sa\big{(}Th_{j}\big{)}\) is the pseudospectral acceleration for \(j\) mode. Results are shown in Table 4. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Period**} & \multirow{2}{*}{\({}^{1}\Gamma_{hj}\)} & \multicolumn{8}{c}{**Pseudospectral Horizontal Acceleration**} \\ \cline{3-10}  & & \({}^{2}\)**Sa-NILT** & \({}^{2}\)**Sa-CALE** & \({}^{2}\)**Sa-JAMI** & \({}^{2}\)**Sa-SCRU** & \({}^{2}\)**Sa-LANE** & \({}^{2}\)**Sa-HMTT** & \({}^{2}\)**Sa-RABO** \\ \hline
0.398 & 0.698 & 0.665 & 0.599 & 0.261 & 0.296 & 0.412 & 0.265 & 0.174 \\
0.108 & 0.196 & 1.864 & 1.059 & 0.613 & 0.532 & 0.548 & 0.348 & 0.309 \\
0.052 & 0.043 & 1.232 & 0.563 & 0.909 & 0.405 & 0.457 & 0.213 & 0.233 \\ \(\Sigma\Gamma_{h}\) = 0.937 & & & & & & & & \\ \(Sa\ H_{SRSS}\) & 0.342 & 0.270 & 0.128 & 0.134 & 0.177 & 0.114 & 0.078 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Accelerations due to horizontal earthquake in the 6-story building. Figure 12: Plastic hinges for earthquake SCRU at the end of the seismic register. The red/blue color in the color contours indicates higher/lower degree of plastic rotation. Likewise, for the vertical accelerations with the highest modal participation (not necessarily the first three as mentioned before) \(Sa\ V_{SRSS}\) is computed with:  
\[Sa\ V_{SRSS}=\sqrt{\sum_{j=1}^{n}\frac{1}{n}\left[\Gamma_{vj}\times Sa\big{(}Tv_{ j}\big{)}\right]^{2}} \tag{2}\]  
where \(\Gamma_{vj}\) stands for the mass vertical modal participation factor for the vertical direction and \(Sa\big{(}Tv_{j}\big{)}\) denotes the pseudospectral accelerations obtained for each spectrum (i.e., NILT, CALE, etc.) in the vertical direction. The values are listed in Table 5.  
To inspect which earthquake ground motion causes more damage when its horizontal and vertical components are simultaneously applied to the 6-story building, the information from Tables 4 and 5 is also included in Table 6. Table 6 contains a new value termed as \(AF\) (adjustment factor) that is calculated as the inverse of the sum of the modal participations \(\Sigma\Gamma_{h}\) and \(\Sigma\Gamma_{v}\), for the modes included in Tables 4 and 5, respectively; these are the inverse of 0.937 and 0.677, leading to 1.066 and 1.477, respectively. Then, \(Sa_{H\&V}\) is computed with:  
\[Sa_{H\&V}=\sqrt{\frac{\Big{(}(\Sigma\Gamma_{h})^{-1}\times(Sa\ H_{SRSS})\Big{)} ^{2}+\Big{(}(\Sigma\Gamma_{v})^{-1}\times(Sa\ V_{SRSS})\Big{)}^{2}}{2}} \tag{3}\]  
which is used to complete the values listed in Table 6.  
The values in Table 6 and the corresponding rotations from the non-linear analyses are depicted in Figure 13 (blue points) for the beams reported in Figures 7 and 8.