## 2 Dynamic Response of a Multi-Story STMF with SMAs
### Dynamic Time History Analysis  
To verify the effectiveness of the SMA incorporation in the STMF configuration, the dynamic response of the conventional STMF (abbreviated as Fr1) is compared with the corresponding one of the proposed configurations with the full diameter of SMAs in the special segment (Fr2) and with the one with half the diameter (Fr3) listed in Table 1.  
In our work, the systems/structures dynamics were analyzed until the ground motion seizes. Thereafter (free motion), the frames' recentering (due to the SMAs) capability will surely exhibit its benefits. At the end of the Conclusion section, the continuation of the analysis is suggested for future work. However, in the very recent literature, other types of self-centering dissipative braces/devices were reported. These can be purposely designed to ensure the absence (or at least minimize) of residual inter-story drifts at the end of the earthquake shaking. In this regard, one must quote the pertinent works of [PERSON] et al. (2015) and [PERSON] et al. (2016). Before discussing the dynamic history analysis results that follow, and to gain a first insight of the dynamics of the frames considered, the outcome of free vibration in terms of the two first eigen periods is in order. These are contained in Table 2.  
The presence of the full diameter of SMAs decreases the fundamental period to about 11% and the second one to about 12.1%. Contrary to this, for Fr2, an increase in \(T_{1}\) and \(T_{2}\) to about 31% and 25%, respectively, is observed. Seven benchmark earthquake records were selected with varying characteristics of PGA and magnitude, as shown in Table 3. These can be considered sufficient in both number and salient features for the dynamic time history analysis given in this section. Their vast response spectrum records, as well as the numerous average response spectra, not only vary but also differ significantly, but the two first periods of the structures dealt with are contained in the period of each spectrum. The direct integration of the equations of motion is accomplished using the numerically dissipative \(\alpha\)-integration algorithm ([PERSON] et al., 1977) or a special case of the former, the well-known Newmark scheme ([PERSON], 1959), with automatic time-step adjustment for optimum accuracy and efficiency; modeling of seismic action is achieved by introducing acceleration loading curves (accelerograms) at the supports. \begin{table}
\begin{tabular}{c c c} \hline
**Floor** & **SMA Diameter (m)** & **\(V_{ne}\) (kN)** \\ \hline
9 & 0.035 & 598 \\
8 & 0.046 & 1019 \\
7 & 0.052 & 1308 \\
6 & 0.057 & 1581 \\
5 & 0.061 & 1821 \\
4 & 0.061 & 1821 \\
3 & 0.061 & 1821 \\
2 & 0.066 & 2126 \\
1 & 0.066 & 2126 \\ \hline \end{tabular}
\end{table} 
Table 1: SMA diameter/diagonal at each floor. \begin{table}
\begin{tabular}{c c c} \hline
**Frame** & **\(T_{1}\) (s)** & **\(T_{2}\) (s)** \\ \hline Fr1 & 3.45 & 1.24 \\ Fr2 & 3.07 & 1.09 \\ Fr3 & 4.52 & 1.55 \\ \hline \end{tabular}
\end{table} 
Table 2: First and second eigen periods of the frames dealt with. For the first earthquake record selected, the Loma Prieta one (LPE), using SMAs as in Table 1 per diagonal, seems to lead to a symmetrical response, as shown by the displacement time history (Figure 2). Recentering is encountered, due to the presence of the SMAs, which is quite important and advantageous for the size of the structure dealt with. SMAs respond linearly at lower floors up to the 5 th floor, and at the 6 th, 7 th, and 8 th floors, they exhibit a small loop, and at the 9 th floor, a loop of 0.057 m (2.1% strain) maximum displacement (Table 4). The incorporation of half the calculated diameter of SMAs per diagonal leads to a more flexible structure (Figure 3). SMAs at lower floors up to the 7 th floor undergo deformations