## 3 Benchmark Highway Bridge  
Since benchmark structures are essential for comparing various control approaches, it was decided that a benchmark structural control model for highway bridges would be developed during the meeting of the ASCE (American Society of Civil Engineering) committee in 2003. For this purpose, bridge 91.5, located in Orange County of Southern California, was selected. This is a continuous, pre-stressed concrete box-girder bridge, with two spans with a length of 58.5 m and two abutments skewed at 33 degrees. The deck is supported by a pre-stressed outrigger with a 31.4-m length and 6.9-m height, resting on two pile groups. The columns are around 6.9 m high. More details about this 4-line bridge are available in the definition paper [18]. A three-dimensional model of this bridge was developed using ABAQUS software. This process included two phases. In the first one, eight seismic isolators were applied to separate the bridge from its end abutments. At this stage, the deck was connected to the central pier (semi-isolated model) ([PERSON] et al. [18]). In the second phase, two isolators were used in the middle piers in addition to the eight aforesaid isolators (completely isolated model) ([PERSON] et al. [26]). The second model was applied in this research, as shown in Figure 2.  
Figure 1: Super-elastic behavior of SMA. In the second model, [PERSON] et al. [26] assumed that the bridge superstructure, including the deck and the beam, has elastic behavior. The soil-structure interaction was considered at the end of the abutments and embankments. This effect was modeled by utilizing the springs and dashpots independent of the frequency, and the base isolators were modeled as shear elements. Therefore, their torsional rigidity and flexural stiffness were initialized to 0, and their vertical axial stiffness initialized to approach infinity. The bearings were idealized by the bi-directional bilinear plasticity model, as described by [PERSON] and [PERSON] [1]. They wrote this model in ABAQUS as a subroutine. The aforesaid model can be mathematically introduced by utilizing Equations (1) and (2). The nonlinear behavior of the isolator in the X-direction is independent of its behavior in the Y-direction. It means that the restoring force in the X-direction is only related to the displacement in direction X. Also, the mass and damping of these bearings are eliminated. In these equations, \(F_{x}\) and \(F_{y}\) are the restoring forces in the X and Y directions, respectively, \(K_{px}\) and \(K_{py}\) are the post-yield stiffness of the isolator in the X and Y directions, respectively, and \(K_{ex}\) and \(K_{ey}\) are the pre-yield stiffness of the isolators in the X and Y directions, respectively. The yield displacements in the X and Y directions are denoted by \(\overline{U}_{x}\) and \(\overline{U}_{y}\), respectively:  
\[F_{x}=K_{px}U_{x}+\big{(}K_{ex}-K_{px}\big{)}\overline{U}_{x}, \tag{1}\]  
\[F_{y}=K_{py}U_{y}+\big{(}K_{ey}-K_{py}\big{)}\overline{U}_{y}. \tag{2}\]  
After developing the three-dimensional model of the bridge with 430 degrees of freedom, [PERSON] et al. [26] calculated the mass, stiffness, and damping matrices. In this process, the nonlinearity effects were considered. The dynamic equation of this nonlinear model is expressed in the form of Equation (3). In this equation, the incremental displacement, ground acceleration (including two horizontal components), incremental controlling force, mass, and stiffness matrices are denoted by \(U\), \(\bar{U}\), \(F\), \(M\), and \(K\), respectively. Note that the stiffness matrix of the structure includes linear and nonlinear sub-matrices. \(\eta\) and \(b\) are the loading vectors for the ground acceleration and control forces, respectively:  
\[M\ddot{U}(t)+C\dot{U}(t)+K(t)U(t)=-M\eta\ddot{U}(g)+bF(t). \tag{3}\]  
Equation (3) can be solved with the help of the Newmark integration tactic. Although MATLAB software has various tools for solving ordinary differential equations, they are not able to find the nonlinear structural responses.