72 O. R. Abuodeh and L. Redmond 00.511.522.53 Time on Bridge (sec) -1.5 -1 -0.5 0 0.5 1 1.5 a b Acceleration (m/s 2 ) HC Bridge Abaqus Bridge 0 0.5 1 1.5 2 2.5 3 Time on Bridge (sec) -6 -4 -2 0 2 4 6 Acceleration (m/s 2 ) HC Vehicle Abaqus Vehicle Fig. 8.5 Half-Car dynamic response curves. (a) Acceleration signal of bridge at midspan. (b) Acceleration signal of vehicle of the vehicle, respectively. It can be observed that the acceleration at midspan of the bridge for both models are not fully in line with each other. This is attributed to the method employed when distributing the contact force of the tire to the adjacent nodes of a beam element. Yang et al. [2] used the cubic Hermitian interpolation function for the transverse displacement of the element to compute the displacement, velocity, and acceleration of two nodes based on the position of the vehicle within contacted element. However, the proposed Abaqus FE model uses a node-to-surface contact formulation in which the contact force is resolved using a hard contact pressure-overclosure relationship and is based on the tire’s contact force as a function of its penetration [11]. This means that Abaqus internally generates a stiffness matrix for the contact area during the analysis in which the equivalent contact force is computed relative to the node of that contacted element, thus causing a smoothing effect on the acceleration response, as shown in Fig. 8.5a. Furthermore, discrete Fourier transform (DFT) is carried out to map acceleration signals into their respective frequency domains for further verification. The computed peak spectrum amplitudes and their corresponding frequencies are 0.0113 m/s2 and 2.20 Hz in both models, respectively, for the vehicle acceleration signals and 0.00880 m/s2 and 1.99 Hz, respectively, for the bridge acceleration signals in both models. Figure 8.5 shows the results that were extracted from the half-car simulation for both the proposed FE model and the hard-coded (HC) FE model [1], where Fig. 8.5a, b are the acceleration signals recorded at midspan of the bridge and center of mass of the vehicle, respectively. As opposed to the quarter-car simulations, the half-car simulation demonstrates bridge acceleration curves that are more in line, as shown in Figs. 8.4a and 8.5a. This is attributed to the effects that material damping and surface roughness have on the dynamic response of the bridge where the sharp amplitudes that are supposed to occur in the HC FE model are smoothened. Similar to the quarter-car simulation, the DFT of the signals are computed where the peak spectral amplitudes and frequencies are 13.7 m/s2 and 3.66 Hz, respectively, for the vehicle signals in both models, respectively, and 0.0947 m/s2 and 6.22 Hz, respectively, for the bridge signals in both models. 8.4 Conclusion This study proposes an efficient framework to construct a VBI model entirely within Abaqus using minimum coding from the user. The implementation of such a framework can be advantageous to users who are interested in exploring the realm of VBI to include nonlinear effects while maximizing computation efficiency. In addition, this study can be expanded to a three-dimensional outlook to increase the user’s parameter space by including more modal shapes when processing the data using physics-based or ML-driven approaches. The presented framework is composed of representing the vehicle and bridge bodies using Abaqus’s built-in element definitions; defining appropriate material definitions that reflect the physical aspects of the problem; employing a node-to-surface contact formulation that is responsible for coupling the vehicle and bridge; and defining the numerical methods used to solve a typical VBI problem. As a result, the proposed approach was successful in producing FE models that agree with the VBI models employed in published literature. The following can be concluded from the present work:
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