# Behaviour Investigation of SMA-Equipped Bar Hysteretic Dampers Using Machine Learning Techniques
## 4 Experimental Specimens and Numerical Models
### Numerical Models
In total, 630 different SMA-BHD models were constructed for which the \(O_{0}\_{L}\_{N}n\) was used as a notation, to label them. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mechanical Property** & **Set 1** & **Set 2** & **Set 3** \\ \hline \(E_{SMA}\) (GPa) & 30 & 24.6 & 28 \\ \(\epsilon_{L}\) (\%) & 4.8 & 4.10 & 4.25 \\ \(\sigma_{r}^{EA}\) (MPa) & 350 & 280 & 320 \\ \(\sigma_{s}^{SA}\) (MPa) & 370 & 350 & 460 \\ \(\sigma_{s}^{AS}\) (MPa) & 150 & 250 & 260 \\ \(\sigma_{f}^{AS}\) (MPa) & 135 & 40 & 190 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of SMA materials. Figure 9: Hysteresis curves of experimental sample ([PERSON] et al., 2016) and numerical model of (**a**) rubber isolator and (**b**) rubber isolator equipped with steel bar hysteretic damper. Here, \(o\) refers to the diameter of the SMA rebar, followed by the calculated length and numbers of SMA bars, \(n\) and \(l\), respectively. A schematic of hysteretic damper models of SMA bars of varying numbers (1 to 10) is presented in Figure 10.  
In each model, hysteresis curves were used to evaluate support reactions (_Force_) versus displacement (_Dis._). Figures 11 and 12 show some examples of hysteresis curves for selected SMA-BHD numerical models with mechanical features of Set 1 and different numbers on bars (with 50 mm diameter) and various cross-sectional diameters (with 10 numbers of bars), respectively. It should be mentioned that the hysteretic curves of some models resulted in just elastic part, which could not obtain any information regarding the post-yield properties. Therefore, from the total number of 630 numerical models, 389 models were selected. Hysteresis curves derived from the numerical models were evaluated to estimate the effects of length, numbers, and the cross-sectional diameter of SMA bars on the cyclic behaviour of SMA-BHDs. As shown in Figure 13, the hysteresis curves of each curve were idealised as bilinear envelope curves. The ideal bilinear curves were derived by means of displacements and forces associated with the first yield point (\(D_{y\_}\)_Hys._, \(F_{y\_}\)_Hys._), and the ultimate loading point (\(D_{u\_}\)_Hys._ and \(F_{u\_}\)_Hys._) of hysteresis curves. This paper identified the initial yield point in the hysteresis curve based on the first slope change, and the ultimate loading point was determined based on the final loading displacement stage \(D_{u\_}\)_Hys._ (0.06 m). A line with a slope equal to the hysteresis curve slope before the first yield point determined the first line of the bilinear curve. The second part of the bilinear curve was depicted based on the equal slope of the hysteresis curve slope before the ultimate point was reached. The bilinear curve had a yield point (\(D_{y\_}\)_Bi_ and \(F_{y\_}\)_Bi) where the lines between the first and second parts intersected. Figure 13 illustrates that because of the nonlinear behaviour of hysteresis curves, the resulting displacement and the corresponding force at the yield point in bilinear curves, \(D_{y\_}\)_Bi_ and \(F_{y\_}\)_Bi, respectively, are greater than the displacement and force indicative of the first yield point in the hysteresis curves (\(D_{y\_}\)_Hys._ and \(F_{y\_}\)_Hys._, respectively). In contrast, considering the same ultimate point, the hysteresis curves and bilinear curves exhibit equal displacement (\(D_{u\_}\)_Hys._ = \(D_{u\_}\)_Bi._ = 0.06 m) and equal force (\(F_{u\_}\)_Hys._ = \(F_{u\_}\)_Bi._) at the ultimate point. Figure 10: Schematic of SMA-BHD numerical models of variable number of bars: (**a**) one; (**b**) two; (**c**) three; (**d**) four; (**e**) five; (**f**) six; (**g**) seven; (**h**) eight; (i) nine; (j) ten. Apart from displacement and forces at the first yield (\(D_{y}\) and \(F_{y}\)) and ultimate (\(D_{u}\) and \(F_{u}\)) points, there are additional parameters in hysteresis and bilinear curves that can help in assessing the cyclic behaviour.