## 2 Dynamic Response of a Multi-Story STMF with SMAs
### An Alternative Design Approach  
Instead of distributing the SMA X-diagonals in the special segment of every floor according to Table 1 and the half diameter of these, we introduce new proposed SMA distributions, based on incorporating the same diameter of SMA X-bars in each special segment in all floors of the frame. Namely, we introduce three new cases: the first, designated as DC1, is related to the incorporation of the mean Table 1 diameter of SMA bars to every floor i.e., bars of the diameter of 0.056 m at each diagonal; the second, DC2, the half of this number, i.e., 0.028 m; and the third, DC3, twice the mean number, i.e., 0.112 m.  
As in Section 2.1, and for the same reasons mentioned therein, modal analysis results are also presented at this point and given in Table 11.  
Figure 7: Comparison of maximum inter-story drifts for the design cases considered: LE. DC1 and DC3 are related to a decrease in both eigen periods. Namely, for DC1, this amounts to 6.38% for \(T_{1}\) (small but not negligible) and 12.9% for \(T_{2}\). DC3 leads to a significant decrease in \(T_{1}\) and \(T_{2}\) up to 23.48% and 25.80%, respectively. However, for DC2, both \(T_{1}\) and \(T_{2}\) exhibit a significant increase (37.68% and 20.97%). For the Loma Prieta earthquake, case DC1 SMAs respond to the limits of superelasticity (Table 12). The maximum inter-story drift diagram (Figure 8a) shows uniform drift distribution. By comparing case DC1 drifts to those of the structure incorporating \(V_{ne}\) dimensioning (Figure 4a), it can be observed that case DC1 drifts are larger on lower floors and smaller on upper floors. This is justified as the diameters obtained from \(V_{ne}\) dimensioning (Table 1) are smaller than those of the DC1 case at upper floors (7 th, 8 th, 9 th floors) and larger at lower floors (1 st to 5 th floors). The incorporation of half the case DC1 diameter of SMAs (case DC2) per diagonal leads to a more flexible structure with the SMAs exhibiting larger strains than the DC1 case, but still within the limits of superelasticity. Despite this, drifts at the upper floors are larger than the 2% target drift limit. Case DC3 SMAs respond elastically at all floors, and this indicates that the input energy is dissipated through the formation of plastic hinges at regions outside the special segment. Even though the special segment is pin-ended, the free rotation of the chord members inside the special segment relative to the trusses on either side is not allowed, due to the stiffness introduced by the large SMA diameter. \begin{table}
\begin{tabular}{c c c} \hline
**Frame** & \(\mathbf{T_{1}\) **(s)** & \(\mathbf{T_{2}\) **(s)** \\ \hline Fr1 & 3.45 & 1.24 \\ DC1 & 3.23 & 1.08 \\ DC2 & 4.75 & 1.50 \\ DC3 & 2.64 & 0.92 \\ \hline \end{tabular}
\end{table} 
Table 11: First and second eigen periods of the frames dealt with in the foregoing approach. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{2}{c}{**DC1**} & \multicolumn{2}{c}{**DC2**} & \multicolumn{2}{c}{**DC3**} \\ \hline
**Floor** & \begin{tabular}{c} **Displacement** \\ **(m)** \\ \end{tabular}  & **Strain** & \ \begin{tabular}{c} **Displacement** \\ **(m)** \\ \end{tabular}  & **Strain** & \begin{tabular}{c} **Displacement** \\ **(m)** \\ \end{tabular}  & **Strain** \\ \hline
9 & 0.015 & 0.55\% & 0.1 & 3.7\% & \begin{tabular}{c} Linear \\ response \\ Linear \\ response \\ response \\ response \\ response \\ response \\ response \\ response \\ response \\ response \\ \end{tabular}  & Negligible \\
6 & 0.04 & 1.4\% & 0.08 & 3.0\% & \begin{tabular}{c} Linear \\ response \\ Negligible \\ Negligible \\ Negligible \\ Negligible \\ Negligible \\ Negligible \\ Negligible \\ Negligible \\ Negligible \\ Negligible \\ Negligible \\ Negligible \\ \hline \end{tabular}  \\
1 & 0.03 & 1.1\% & 0.05 & 1.8\% &