# Effectiveness of LRB in Curved Bridge Isolation:
## 2 Modeling  
A horizontally curved reinforced concrete bridge (Figure 2) (with a radius of curvature of 315 m) has been considered for the study ([PERSON] et al.) [29]. The bridge consists of 7 spans that comprise 5 intermediate sections of 25 m in length and a final section of 20 m. The total length of the bridge is 165 m. The superstructure consists of a single chamber box girder of a constant section with an area of 3.099 m\({}^{2}\) and a longitudinal bending moment of inertia of 0.599 m\({}^{4}\). M40-grade concrete has been considered. All the piers are made of M30 grade concrete, with a height of 11 m and a circular cross-sectional area of 1.767 m\({}^{2}\). Spiral reinforcement (20 \(\Phi\)) is used as stirrups with a pitch of 10 cm, and longitudinal reinforcement bars of 15 nos. 28 \(\Phi\) are used together with a cover layer thickness of 5 cm. The bridge was modeled using the finite element program in SAP2000 [30] software. The bridge superstructure and piers were modeled using beam elements, and masses were concentrated at discrete points. Figure 2: Curved bridge considered for the study. The pairs are considered to be fixed as they are considered to be resting on the rock. The abutment is assumed to be rigid. LRBs have been modeled as link elements (Figure 3). The force-deformation behavior of LRB is shown in Figure 4a,b. The bridge is supported by roller bearings at the abutment and fixed to the bottom of the pier. For each case, two LRBs are considered at each abutment location, and four LRBs are considered at each pier location. In the model, the global longitudinal (X) axis is the straight line connecting two abutments, and the global transverse (Y) axis is the perpendicular horizontal direction with respect to the X axis. The vertical axis is the global Z direction. For gravity loads, static analysis is typically adopted. Non-Linear Time History Analysis (NLTHA) was used to analyze the seismic loads of bridge structures with different types of bearings. Since the bridge is curved, the resultant of the responses in the X and Y directions were considered. The lumped plasticity model was considered in the study. In the finite element model, the mass is concentrated at discrete points between segments (elements) automatically generated by SAP2000 [30]. The SAP2000 program automatically considers mass distribution. The program automatically calculates the translational mass of all elements in the global direction of the bridge and assigns them as the concentrated mass at each node according to the length of the tributaries. In order to approximate the distribution of mass with concentrated mass, it is necessary to define a sufficient number of nodes and segments (units). According to ATC 32 [31], it is recommended that each superstructure span has at least 4 segments and piers have at least 3 segments. In this study, 11 types of elements have been considered for piers. The superstructure (deck) spans have been modeled by 20 elements for a 20-m span and 25 elements for a 25-m span. Table 1 mentions  
Figure 4: Lead rubber bearing (**a**) analytical model, (**b**) force-deformation behavior. Figure 3: Modeling of isolation bearings, pier, and bridge deck junction. the details of the elements considered in each part of the bridge. Non-linearity in the pier was considered by plastic hinges, and the hinge properties were considered using a concentrated plastic hinge model according to FEMA 356 [32]. The Bouc-Wen model [33] was used in this study and is shown in Figure 4. The model is presented by three parameters: (a) Post Yield Stiffness ratio \(k_{2}\) or \(k_{d}\); (b) Characteristic strength \(Q_{d}\) inter concept of zero displacements; (c) Yield strength \(f_{py}\). The Characteristic Strength \(Q_{d}\) of LRB is provided by the yield strength of lead in shear \(f_{py}\) and cross-sectional area of the lead plug \(A_{p}\), as given by the following equation.