68 O. R. Abuodeh and L. Redmond fidelity representations of both the bridge and vehicle while employing a built-in contact formulation through a commercial FE environment alone [12–15]. While these unique frameworks offer accurate and reliable results, there exists a trade-off between frameworks that require sophisticated coding of the user but result in high computational speed (1 and 2) and frameworks with simple implementation but longer computation times (3). For instance, Yang et al. [7] manually coded in the VBI system for a sprung mass traveling across Euler beam elements to study the feasibility of extracting the bridge’s natural frequency from the acceleration data measured from the sprung mass. However, the VBI system used does not account for nonlinear geometry or inelastic material definitions. In particular, Yu et al. [10] sought to overcome these shortcomings by linking MATLAB to Ansys, a commercial FE software with a suite of material definitions, and independently solving the equations of motion of the vehicle to obtain time histories of the tire locations and their respective contact forces. The time histories of these contact forces are applied to the bridge in a separate transient analysis using Ansys. Once the analysis is completed, the resulting time histories of the bridge nodal displacements are applied back to the vehicle model in MATLAB to solve the equations of motion of the vehicle and recompute the new tire forces that begin the analysis cycle again. This process continues until the difference between the tire forces computed in MATLAB and Ansys are minimal. The previous framework is limited to a specialized audience who is proficient in automating FE models with separate programming languages. In addition, it requires multiple iterations for the model to converge for a single vehicular trip, which makes it undesirable for applications like FE model updating in DBHM with ML algorithms. Developing a VBI system entirely within a user-friendly commercial FE software would simplify the procedure and can be useful to a broader audience. However, to the authors’ knowledge, the available published articles that analyze VBI systems completely in a commercial FE software are too computationally expensive for use in physics-based ML algorithms for DBHM [12–15]. For instance, Kwasniewski et al. [15] carried out an extensive 3D FE model of a VBI system involving a heavy truck and a selected highway bridge in Florida within LS-DYNA. The bridge deck, girders, steel reinforcement, and prestressed strands were all included in the model using elastic material definitions. The truck was completely replicated within the FE environment such that the tire was modelled using shell elements with two layers, an elastic rubber material and a fabric material for tire cord, while also employing an airbag option that simulates internal pressure in all tires. As a result, approximately 420,000 elements were generated for this study, including multiple point constraints and contact algorithms used during the dynamic analysis. The researchers reported a good correlation between field measurements and FE analysis. However, fully replicating a field test can prove to be time-consuming and difficult since appropriate modeling strategies must be followed to prevent instabilities that result from nodal misalignment. The present study attempts to bridge the aforementioned gaps by completely proposing a framework to construct an efficient VBI entirely within a commercial FE software (Abaqus). A quarter-car and half-car models are completely modeled in Abaqus where the VBI is defined using a robust node-to-surface interaction command. Bridge/vehicle data reported in published articles [1, 7] are used to verify the proposed modeling framework. 8.2 Model Development The most common method for developing a VBI model is to model the vehicle as a sprung mass, hereafter referred to as a quarter-car model, with two degrees of freedom; vertical motion of the center of mass and tire point of the vehicle model. An extension of the quarter-car model that accounts for additional modes is the half-car model, which uses the bicycle concept and has six degrees of freedom, two of which are vertical bounce and pitching motion of the half-car body and the remaining four are vertical motions of the front wheel, rear wheel, front tire, and rear tire. Both of these models are summarized in Fig. 8.1, where the subscripts “q,” “v,” “t,” “R,” “F,” “S,” and “R” are the quarter-car, half-car, tire, rear, front, rear, sprung, and un-sprung, respectively, and u and θ are vertical and pitch degrees of freedom, respectively. Mand I are the mass and mass moment of inertia of the vehicles, respectively. K, C, andNare the stiffness, damping, and contact node of the vehicles, respectively. For the bridge model, Eb is the elastic modulus of the beam element, Ib is the moment of inertia of the beam element, Ab is the area of the beam element, μb is the mass per length of the beam element, and V is the constant velocity the vehicle drives. In Abaqus, the vehicle and bridge bodies are modelled independently, and contact is represented using one of Abaqus’s interaction definitions. Finally, the vehicle body is pushed with a user-given displacement across the bridge within a time period that is equal to the user-given velocity, while gravity is being applied to the mass of the vehicle. The bridge elements are discretized with 2-node B23 Euler beam elements which use cubic interpolation functions with consistent mass matrix formulations. The quarter-car body is discretized with 4-node CPS4R plane stress elements and prescribed a rigid body definition. The half-car body is discretized using 2-node B21 linear Timoshenko elements which
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