## 3 Numerical Study
### Finite Element Model  
The benchmark model was constructed based on the experimental test of a double-pier RC bridge conducted by [PERSON] et al. [19], of which details are represented in Figure 7a. The column, cap beam, foundation and steel panel zone were modeled using eight-node reduced integration brick elements (C3D8R), of which the mesh sizes were from 6 to 80 mm based on the mesh convergence test. The truss T3D2 element was employed to define the longitudinal rebars, stirrups and BRB. The 'concrete damaged plasticity' material model was used for the concrete, and the bilinear material model was used for all longitudinal rebars, stirrups, steels and BRBs.  
C40 concrete with a compressive strength of 41.3 MPa was used for the foundation, the columns and the cap beam. HRB335 reinforcement bars with a yielding strength of 392 MPa were adopted for the longitudinal rebars and the stirrups. The bolts were made of S690Q high-strength steel [18], and Q345 and Q235 steel were selected for the steel panel zone and the BRB, respectively, of which the main mechanical indexes are represented in Table 1. The interaction between the rebars and concrete was modeled using the embedded constraint prescribed in ABAQUS, while the bond-slip behavior between them was ignored. The coupling constraint was adopted between the BRB and gusset plate, and the other connections were modeled using the tie constraint.  
With respect to the boundary conditions, the foundation was fully fixed on the ground. To account for the weight of the superstructure, a vertical load corresponding to 20% of the column's bearing capacity was applied on the top of the cap beam. The lateral cyclic displacement excitation was loaded at the left end of the cap beam. As shown in Figure 8, the imposed story drift was 1/1200, 1/550, 1/200, 1/100, 1/50 and 3/100 [30]. In this  
Figure 7: The benchmark model: (**a**) Details of the RC double-column bridge; (**b**) FE model of bare bridge bent; (**c**) FE model of BRB-RCB.  
study, strength degradation caused by the repeated cyclic load was not the issue of concern, and, thus, only one cycle was carried out at each displacement amplitude. The eigenvalue analysis was initially conducted to obtain the buckling mode, and then the scaled buckling mode from the buckling analysis was imposed on the model as the initial imperfection. Based on [PERSON] et al.'s research [31], a maximum imperfection of 0.01\(t_{\text{g}}\) was considered in the subsequent nonlinear analysis.  
Figure 9 shows the comparison of the hysteretic curve of the double-pier RC bridge model with the test result [19], as well as the comparison of both backbone curves depicted in Figure 10, respectively. From the comparisons of the hysteretic and backbone curves, it can be found that the finite element model accurately predicts the loading stiffness, unloading stiffness and hysteretic behavior using the material properties mentioned above. The results of the parametric study are summarized in Table 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **Elastic Modulus (GPa)** & **Yield Strength (MPa)** & **Ultimate Strength (MPa)** & **Poisson’s** **Ratio** \\ \hline Q235 & 210 & 235 & 428 & 0.3 \\ Q345 & 210 & 345 & 540 & 0.3 \\ S690Q & 210 & 794 & 856 & 0.3 \\ HRB335 & 210 & 392 & 498 & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mechanical properties of materials [18; 19].  
Figure 8: Loading protocol.  
Figure 9: Numerical vs. experimental hysteretic curves.