## 3 FE Study
### FEM of BRBs  
The BRB2, BRB4, and BRB7 had the same CP, but their CMs were different [2]. The additional end restrainer of BRB2 was formed by welding 20 \(\times\) 10 steel plates on the alignment of SHS 100 \(\times\) 3. The tied constraint was used to simulate the welded connection between their surfaces. It was also used between the steel tubes and concrete for FEM of all BRBs because no slip was observed between them during the test. The CM of the BRB7 consisted of reinforced concrete. The reinforcements were modeled using two node truss elements (T3D2). The truss elements were joined to the concrete solid element using embedded constraints. The reference points were connected with the surfaces at both ends of the BRBs using a continuum-distributing coupling constraint. The boundary conditions were defined in these reference points. The FEM of all BRBs is given in Figure 10.  
The stress-strain curve of steel material under cyclic loading is different from that under monotonic loading [63]. Therefore, the combined isotropic-kinematic hardening material model was used for the behavior of the CP under cyclic loading. The hardening parameters of steel material should be calibrated from the cyclic coupon test data [39]. In this study, the fatigue tests of coupon specimens for CPs were not available, but tensile coupon tests of material for CPs were conducted. Therefore, to obtain the hardening parameters of steel material under the cyclic loading, a numerical parametric study was carried out on the BRB2 using combined isotropic-kinematic hardening steel material models of many previous studies in the literature [36; 46]. In addition, these parameters of material models were suitably calibrated by the trial-and-error method to match the numerical and test results. The calibrated material models were then used in all FE analyses. All parameters for material models are given in Table 2. The other components (concrete, steel tubes, and steel plates) remained within the elastic limit during the tests [2]; so, they were taken as elastic in the FE analyses. The mechanical properties of these components  
Figure 9: The contact pairs of slip-critical connections. are given in Table 3. Numerical parametric studies were performed on the BRBs to find the effects of the material models of steel used for CP and the value of the friction coefficient between the CP and CM. The results of these parametric studies are given in Section 4. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multicolumn{8}{c}{**Kinematic Hardening Parameters (Mpa)**} & \multicolumn{8}{c}{**Isotropic Hardening Parameters (Mpa)**} \\ \cline{2-13}  & \(\sigma_{0}\) & \(C_{1}\) & \(\gamma_{1}\) & \(C_{2}\) & \(\gamma_{2}\) & \(C_{3}\) & \(\gamma_{3}\) & \(C_{4}\) & \(\gamma_{4}\) & \(C_{5}\) & \(\gamma_{5}\) & \(Q_{\infty}\) & \(b\) \\ \hline Korzekwa and Tremblay [46] & 270 & 8000 & 75 & - & - & - & - & - & - & - & 110 & 4 \\ [PERSON] et al. [36] & 270 & 95,000 & 1300 & 40,500 & 680 & 5000 & 120 & 500 & 2.5 & 200 & 2 & - & - \\ Calibrated Korzekwa and Tremblay [46] & 270 & 8000 & 75 & - & - & - & - & - & - & - & 110 & 2 \\ Calibrated [PERSON] et al. [36] & 270 & 87,000 & 1300 & 32,500 & 680 & 4600 & 120 & 275 & 2.5 & 160 & 2 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Components**} & \multicolumn{2}{c}{**Young’s Modulus**} & \multicolumn{2}{c}{**Poisson’s**} & \multicolumn{2}{c}{**F\({}_{y}\)**} & \multicolumn{2}{c}{**F\({}_{u}\)**} & \multicolumn{2}{c}{**f\({}_{c}\)**} \\  & \multicolumn{2}{c}{**(MPa)**} & \multicolumn{2}{c}{**Ratio**} & \multicolumn{2}{c}{**(MPa)**} & \multicolumn{2}{c}{**(MPa)**} & \multicolumn{2}{c}{**(MPa)**} \\ \hline CP & 210,000 & 0.3 & 270 & 401 & - \\ SHS 250X6 & 210,000 & 0.3 & 235 & - & - \\ SHS 100X3 & 210,000 & 0.3 & 235 & - & - \\ Concrete & 25,000 & 0.2 & - & - & - & 25 \\ \hline \hline \end{tabular}
\end{table}