## 4 Case-Study Building
### Modal-DBD Design Steps
#### 4.2.10 Modal-DBD Application and Results  
The mass ratios (\(\beta_{m}\)) were obtained by the analysis of the examined Blockhaus system subjected to self-weight gravity loads and superimposed dead loads. More in detail, the mass ratios for the first, second and third floors (relative to the first one) were estimated in 1.0, 0.90 and 0.53, respectively. In accordance with [34] the initial \(\beta_{k}\) values (relative to the first floor) were then estimated for each story by calculating the total available shear-wall length in the selected direction. In the X-direction, in particular, these stiffness ratios resulted in 1.0, 0.79 and 0.79 for the first, second and third level, respectively. In the Y-direction, constant stiffness ratios equal to 1.0 were obtained for the first, second and third floor. The analyses were thus carried out for both X- and Y-directions, starting from the CP level and downwards. The iterative calculation process was carried out as follows:  
* First, an initial estimation of required stiffness values and drift levels was performed;
* The actual stiffness contributions \(\beta_{k}\) of each log-wall partition (for a given drift level) were then taken into account;
* Rounded drift have been evaluated from 0.05% up to 4% at intervals of 0.025, the initial estimated drift are rounded to the closet value in the belonging range. * As a final step, a new normalized modal analysis was performed (using the actual \(\beta_{k}\) values), in order to obtain new story drift estimates and a corresponding set of required equivalent stiffnesses;
* For a seismic performance assessment of the case-study building, the actual-to-required stiffness ratio was calculated for each limit state, log-wall and story level. The overall calculation process was carried out accounting for the stiffness values obtained from the EP model of each log-wall partition (i.e., Table 3). Major calculation outcomes are reported in Tables 4-9, grouped by limit state and (X, Y) direction. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Initial Estimation**} & \multicolumn{4}{c}{**Sbase-Wall Assessment**} & \multicolumn{4}{c}{**Design Verification**} \\ \hline \multirow{4}{*}{**CP Level**} & \multirow{2}{*}{**Story**} & \multirow{2}{*}{**Initial**} & \multirow{2}{*}{**Data**} & \multirow{2}{*}{**Regized**} & \multirow{2}{*}{**Rounded**} & \multicolumn{4}{c}{**Actual \(\mathbf{x_{p}}\)/A(Num)**} & \multirow{2}{*}{**Actual**} & \multirow{2}{*}{**Diff**} & \multirow{2}{*}{**Regized \(\mathbf{x_{p}}\)**} & \multirow{2}{*}{**Actual**} \\  & & \(\mathbf{f_{n}}\) & (\(\mathbf{c_{3}}\)) & \(\mathbf{x_{p}}\) (\(\mathbf{NNum}\)) & & & & & & & & & & & \\ \cline{1-1} \cline{6-13}  & 1 & 1 & 2.50 & 2.37 & 2.50 & 2.36 & CF & 0.82 & CF & 3.21 & 1.20 & 2.50 & 2.20 & 1.40 \\  & 2 & 1.09 & 1.44 & 2.58 & 1.75 & 2.3 & FF & 1.04 & FF & 3.34 & 1.04 & 1.73 & 2.38 & 1.40 \\  & 3 & 1.52 & 0.57 & 3.61 & 0.50 & 2.48 & SF & 1.61 & SF & 4.09 & 1.27 & 0.70 & 2.91 & 1.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Immediate Occupancy (IO) level analysis (\(\chi\)-direction), accounting for the EP shear-wall stiffnesses.