## 2 Dynamic Response of a Multi-Story STMF with SMAs
### Dynamic Time History Analysis
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Table 1 SMAs**} & \multicolumn{2}{c}{**Half the Diameter of Table 1**} \\ \hline \multirow{2}{*}{**Floor**} & **Displacement** & **Corresponding** & **Displacement** & **Corresponding** \\  & **(m)** & **Strain** & **(m)** & **Strain** \\ \hline
9 & 0.057 & 2.1\% & 0.26 & 9.6\% \\
8 & 0.045 & 1.6\% & 0.17 & 6.3\% \\
7 & 0.040 & 1.5\% & 0.10 & 3.7\% \\
6 & 0.035 & 1.3\% & 0.06 & 2.2\% \\
5 & Linear response & Negligible & 0.05 & 1.8\% \\
4 & Linear response & Negligible & 0.05 & 1.8\% \\
3 & Linear response & Negligible & 0.05 & 1.8\% \\
2 & Linear response & Negligible & 0.04 & 1.5\% \\
1 & Linear response & Negligible & 0.032 & 1.2\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum SMA displacements for LPE. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Earthquake/Year/Location** & **PGA (g)** & **Magnitude** & **Horizontal** \\  & & **(Mw)** & **Component** \\ \hline Loma Prieta/1989/California & 0.37 & 6.93 & ESE \\ Northridge/1994/California & 0.57 & 6.69 & NNW \\ Kocaeli/1999/Izmit, Turkey & 0.35 & 7.51 & WSW \\ Imperial Valley/1997/Mexico & 0.32 & 6.53 & NNW \\ Kobe/1995/Japan & 0.35 & 6.90 & NW-WNW \\ Chichi/1999/Taiwan & 0.36 & 7.62 & N-NNW \\ Landers/1992/California & 0.78 & 7.28 & NNE \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of benchmark earthquakes. Figure 2: LPE: (**a**) comparison of the response of the conventional and the proposed STMF (with the full SMA diameter per diagonal); (**b**) 9 th floor damper force–displacement curve (with the full SMA diameter per diagonal). up to 0.1 m (3.7% strain). At upper floors (8 th, 9 th), they undergo deformations up to 0.26 m. This corresponds to a 9.6% strain. For this study, to consider that the SMA deforms in the range of superelasticity, the strain that it undergoes must be under 6%. Consequently, this is a large strain for the SMAs to accommodate and can provoke plastic deformation by slip. As far as inter-story drifts are concerned (Figure 4a), the incorporation of SMAs according to \(V_{ne}\) results in an even distribution with larger values compared to those of the structure with the conventional STMF at the lower floors, and smaller at the upper floors. This can be attributed to the fact that the structure with SMAs has pin-ended special segments and to the fact that SMAs respond linearly at the lower floors, while at the upper floors, they exhibit a loop. The structure with the conventional STMF and the structure with half the calculated diameter of SMAs presents large drift values at upper floors indicating damages. Using a target drift of 2% for a ground motion hazard level, with a 10% probability of exceedance in 50 years as a criterion, as [PERSON] and [PERSON] also used in their 2006 report (see [2]), it is observed that only the structure with the SMAs according to \(V_{ne}\) has drifts below the drift limit. Figure 4: Comparison of maximum inter-story drifts for the design cases considered: (**a**) LPE; (**b**) NE. Figure 3: LPE: (**a**) comparison of the response of the conventional and the proposed STMF (with half the SMAs diameter per diagonal); (**b**) 9 th floor damper force–displacement curve STMF (with half the SMAs diameter per diagonal). For the Northridge earthquake (NE), SMAs deform in the limits of superelasticity up to the 8 th floor (Table 5). On the 9 th floor, the maximum SMA displacement of 0.15 m slightly exceeds the 6% strain limit of superelasticity. The incorporation of half the calculated diameter leads to larger displacements but within the limits of superelasticity, except for the 9 th floor where the SMA displacement grows dramatically up to 0.28 m, corresponding to a 10.3% strain. The performance of all three structures is revealed via the maximum inter-story drift diagram (Figure 4b), where the upper floors exhibit values greater than the target drift limit of 2%.