# Behaviour Investigation of SMA-Equipped Bar Hysteretic Dampers Using Machine Learning Techniques
## 4 Experimental Specimens and Numerical Models
### Numerical Models
Parameters such as elastic stiffness (\(K_{e}\)), post-yield stiffness (\(K_{p}\)), and \(\alpha\), which is the post-yield stiffness ratio and equals to the ratio of post-yield stiffness (\(K_{p}\)) to the elastic stiffness (\(K_{e}\)), could be calculated by Equations (4)-(6):  
\[K_{e}\_Hys.=K_{e}\_Bi.=\frac{F_{y}\_Hys.}{D_{y}\_Hys.} \tag{4}\]  
\[K_{p}\_Hys.=\frac{F_{u}\_Hys.-F_{y}\_Hys.}{D_{u}\_Hys.-D_{y}\_Hys.},K_{p}\_Bi.= \frac{F_{u}\_Bi.-F_{y}\_Bi.}{D_{u}\_Bi.-D_{y}\_Bi.} \tag{5}\]  
\[\alpha\_Hys.=\frac{K_{p}\_Hys.}{K_{e}\_Hys.},\alpha\_Bi.=\frac{K_{p}\_Bi.}{K_{e }\_Bi.} \tag{6}\]  
Figure 11: Typical hysteresis curves of SMA-BHD numerical models with 50 mm diameter (\(\Phi\)50) and different number of bars in Set 1: (**a**) one; (**b**) two; (**c**) three; (**d**) four; (**e**) five; (**f**) six; (**g**) seven; (**h**) eight; (**i**) nine; (**j**) ten. The amount of dissipated energy in dampers is another important parameter that should be taken into consideration. Based on the adopted bilinear curve presented in Figure 14, the process of estimating dissipated energy (\(E\)) in SMA-BHD dampers are reported in Equations (7)-(9):  
\[F_{y\_Bi.}=K_{e\_Bi.}\times D_{y\_Bi.} \tag{7}\]  
\[F_{u\_Bi.}=F_{y\_Bi.}+K_{p\_Bi.}\times(D_{u\_Bi.}-D_{y\_Bi.}) \tag{8}\]  
Figure 9:0  
where \(F_{y\_Bi. }\) and \(F_{u\_Bi. }\) represent the yield and ultimate force, and \(D_{y\_Bi. }\) and \(D_{u\_Bi. }\) show their corresponding displacements in the idealised bilinear curves, respectively. Moreover, \(K_{e\_Bi. }\) and \(K_{p\_Bi. }\) are, respectively, elastic and post-yield stiffness. \(Q\_Bi.\) shows the yield force corresponding to the unloading step of the bilinear curve at the ultimate displacement and can be obtained as \((\sigma/^{AS}\ /\ \sigma/^{EA})\times F_{y\_Bi.}+K_{p\_Bi.}\times(D_{u\_Bi.}-D_{y\_Bi. })\), where \(\sigma/^{EA}\) is the stress relating to the beginning transformation of the austenitic into the martensitic phase, and \(\sigma/^{AS}\) is the stress relating to the ending of the unloading step, as presented in Figure 9:1 Both \(\sigma/^{EA}\) and \(\sigma/^{AS}\) are the mechanical properties of SMA materials (Figure 2) and, therefore, are known parameters. Moreover, \(F_{y\_Bi. }\) is equal to \(K_{e\_Bi. }\times D_{y\_Bi. }\) as presented in Equation (7), where \(K_{e\_Bi. }\) (elastic stiffness of bilinear curve) is equal to \(K_{e\_Hys. }\) (elastic stuffiness of hysteretic curve), as stated in Equation (4), and \(D_{y\_Bi. }\) should be determined as an unknown parameter. Furthermore, as presented in Equation (6), \(K_{p\_Bi. }\) is equal to \(\alpha\times K_{e\_Bi. }\), where \(K_{e\_Bi. }\) is a known parameter (is equal to \(K_{e\_Hys. }\)), and \(\alpha\) should be determined as an unknown parameter. As the ultimate point in bilinear curves is considered to be the same as hysteresis curves, their corresponding displacements are equal (\(D_{u\_Bi.}=D_{u\_Hys. }\)), and therefore, \(D_{u\_Bi. }\) is a known parameter. As a result, as shown in Figure 15, the only unknown parameters in the process of \(Q\_Bi.\) determination are \(D_{y\_Bi. }\) and \(\alpha\) which should be taken to consideration. Instead of utilising hysteretic curves parameters, there are already existing analytical equations which used geometrical and mechanical properties of hysteretic bar dampers to identify their cyclic behaviour. The yield displacement (\(D_{y}\)) and an elastic stiffness (\(K_{e}\)) of SMA-BHDs could be described as follows [58]:  
Figure 9:2  
Figure 9:3  
In Equations (10) and (11), \(f_{y}\) and \(E\) are, respectively, the yield stress and elasticity modulus of materials, and as reported before, \(N\), \(D\), and \(L\) are the numbers, diameter, and calculated length of SMA bars, respectively. In most analytical cases, the ideal bilinear curve, obtained from an experimental hysteretic curve, is considered as the analysis reference.