## 2 Materials and Methods  
### Node-Independent Model for Reinforced Concrete  
When modeling reinforced concrete barriers using the finite element method, the most intuitive approach is to share nodes for the reinforcement and concrete elements. Concrete is specifically modeled using solid elements, and reinforcement using beam elements. Reinforcement can also be modeled with truss elements. If bending is minimal and axial behavior is predominant, truss elements are sufficient; however, beam elements are preferable for capturing the bending or shear behavior of the reinforcement. Figure 1 illustrates an example of modeling reinforced concrete using solid and beam elements. To represent the interaction between reinforcement and concrete, nodes are simply shared without additional constraints. It is important to note that the degrees of freedom for nodes used in solid elements differ from those in beam elements; so, only translational degrees of freedom are shared. In Figure 1, the points marked in purple indicate the nodes connecting the beam and solid elements.  
Both the bridge deck and the barriers installed on the bridge feature a complex arrangement of a significant amount of reinforcement in both longitudinal and transverse directions. Thus, considerable time is required for mesh design to share the nodes of beam elements representing reinforcement and solid elements representing concrete, making the finite element discretization process quite cumbersome. Additionally, in such processes, the aspect ratio of concrete elements may deteriorate to satisfy node sharing. This poor aspect ratio can reduce the accuracy of the numerical solution. To improve the aspect ratio,  
Figure 1: Node-sharing method to connect solid elements and beam elements  
a very fine mesh design is necessary, which increases computational costs and results in a large number of elements. To address these issues, a node-independent model that allows for interaction without matching the nodes of different elements is required. In this study, the node-independent model using the CBIS (Constrained Beam In Solid) model provided by LS-DYNA [24; 27] is applied to reinforced concrete barriers. In the node-independent model, nodes of elements corresponding to reinforcement and concrete do not need to match. Reinforcement and concrete are modeled as beams and solids, respectively. The beam mesh is submerged in the solid mesh, and each possesses independent motion. The node-independent model utilizes a master-slave concept to define the interaction between solid and beam elements. The master for concrete regions is responsible for calculating strains, stresses, and displacements. The slave for reinforcement derives its strains from the master and calculates its stresses accordingly. In the current analysis, both velocity and acceleration need to be constrained to ensure momentum conservation and force balance. The detailed process is as follows: In the first step, beam nodes distribute their nodal mass (\(m\)), velocity (\(v\)), and momentum (\(mv\)) to solid nodes (\(M\), \(V\), \(MV\)). Next, the master nodal velocity (\(V^{\prime}\)) is calculated by dividing the new momentum by the new mass. Finally, the velocity (\(v^{\prime}\)) interpolated by suitable functions \(N\) is assigned to the slave nodes. The slave nodes can move exactly the same way as master nodes. This process is shown in Figure 2. Each beam node is known to have not only its nodal mass derived from the mesh discretization process but also its nodal velocity. However, artificially generated coupling points lack these properties. Therefore, the velocity at a coupling point should be interpolated from the beam end nodes.