8 A Framework for Developing Efficient Vehicle-Bridge Interaction Models Within a Commercial Finite Element Software 69 Fig. 8.1 A schematic of a Quarter-car model on a simply supported beam use lumped mass formulations and are also prescribed a rigid body definition. The stiffness of the quarter-car and half-car models are modelled using SPRING2 elements, and the damping of the half-car model are modelled using DASHPOT2 elements [16]. Essentially, the bridge and vehicle elements are assigned density values to obtain desired inertial properties for the dynamic analysis. For defining density in a vehicle body, if the user desires the vehicle’s center of mass to be located away from its midpoint, uniform density cannot be prescribed. Instead, the beam elements should be partitioned into multiple segments each with a user-defined density that would cumulatively yield the desired total mass, mass moment of inertia, and center of mass location. For a half-car model with its center of gravity at the midpoint between the wheels, it is important to note that more than one element should be used, since the mass moment of inertia calculated from the user-given density can deviate from the theoretical one for courser meshes [16]. In addition, depending on the vehicle configuration employed, the nodes that connect to the spring elements should be defined as either pin or tie nodes, where the pin nodes have only translational degrees of freedom associated with the rigid body (quarter-car model) and the tie nodes have translational and rotational degrees of freedom associated with the rigid body (half-car model) [16]. Furthermore, the vehicle and bridge bodies are coupled together using the contact pair formulation. In this study, nodeto-surface contact pairs are employed where the tire node is the slave node and the beam surface is the master surface. The “hard” contact pressure-overclosure relationship is used to minimize penetration and avoid excessive contact chatter during the analysis. Similar contact formulations were used in a previous work [17], where Yao et al. constructed a framework for including surface roughness in a commercial FE software and verified it only against a quarter-car model. Prior to analyzing the problem, the type and number of numerical methods, hereafter referred to as steps, should be defined. The vehicle needs to be in vertical static equilibrium before traveling across the beam elements. This requires the user to define gravity for the vehicle throughout all the steps. However, placing a *DASHPOT element within the initial static step results in long convergence times to reach static equilibrium. Therefore, the “*Model change” command [16] is used to temporarily remove the dampers during this step followed by adding them back in the next step. The model consists of three steps: two static steps and one implicit dynamic step. The first static step is defined to settle the vehicle body to its static equilibrium position vertically using gravity while removing DASHPOT2 elements. Afterward, the next static step is defined to add back the DASHPOT2 elements while maintaining static equilibrium. Finally, the last implicit dynamic step is defined to push the vehicle with a user-given velocity. During all the steps, the gravity load is only applied to the vehicle body of interest using the “*Dload” [16]. Figure 8.2 shows the “*Dload” and “*Model change” commands that are used in this study where their description can be found in [16]. It is worth noting that the first two steps are not required for a vehicle body without any dampers, similar to the work described in [17]. The Hilber-Hughes-Taylor time integration scheme is used during the implicit dynamic step where the integration parameters α, β, andγare used for the direct integration of the equations of motion [16]. The βandγintegration parameters are part of the Newmark family where values of 1/4 and 1/2 follow the trapezoidal rule, which is numerically non-dissipative
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