# A Vehicle-Bridge Interaction Element: Implementation in ABAQUS and Verification
## 1 Introduction
[PERSON] [31] also proposed mathematical models for the vibration of double-walled carbon nanotubes (DWCNTs) subjected to a moving nanoparticle, employing nonlocal classical and shear deformable beam theories. The study assumed the van der Waals (vdW) interaction force between adjacent CNTs as a linear function of the deflection fields of the innermost and outermost CNTs. Mathematical modeling and analytical solutions were conducted, expressing critical velocities associated with nonlocal beam theories in terms of small-scale effect parameters, DWCNT geometry, and material properties. The study's conclusions revealed that the disappearance of the vdW interaction force and the small-scale effect parameter from a slender DWCNT led to predicted critical velocities approaching those of the innermost carbon nanotube based on classical continuum theory. These findings underscored the significance of considering nonlocal effects and interaction forces when analyzing nanotube structures under the influence of moving nanoparticles. The Euler-Bernoulli beam theory assumes that the cross-sections remain perpendicular to the beam axis during deformation, indicating that the shear strain is negligible. However, in reality, shear deformation can significantly affect the behavior of beams, especially when dealing with slender or highly flexible structures. In such cases, higher-order theories such as the Timoshenko beam theory or shear deformable beam theories should be taken into consideration, which accounts for shear deformation effects more accurately. These higher-order theories provide more accurate predictions for beams with varying slenderness ratios and are better suited for analyzing structures where shear deformation plays a significant role. [PERSON] [32] investigated the analytical solution of the dynamic response of a Timoshenko beam resting on a Winkler-type elastic foundation subjected to a moving concentrated mass, with the possible separation of the moving mass from the beam considered. [PERSON] et al. [33] proposed a discrete element analysis of the dynamic response of Timoshenko beams under moving mass. [PERSON] et al. [34] conducted the finite-element analysis for a Timoshenko beam subjected to a moving mass. [PERSON] [35] analyzed the dynamic response of a functionally graded Timoshenko beam on two-parameter elastic foundations due to a variable velocity moving mass. [PERSON] et al. [36, 37] conducted parametric studies to predict the capabilities of classical and shear deformable beam models excited by a moving mass. The beam slenderness, mass weight, and velocity of the moving mass are introduced as well as various boundary conditions. They defined the critical beam slenderness in which, for slenderness lower than it, the application of Euler-Bernoulli or even Timoshenko beam theories would underestimate the real dynamic response of the system. [PERSON] and [PERSON] [38, 39] also extended the beam theory at the nanoscale. Previous studies have employed various numerical methods such as the Newmark and Runge-Kutta schemes to solve the governing equations for vehicle-bridge interaction problems [40, 41, 42]. These methods often involve an iterative procedure within the time-stepping process, as the vehicle and bridge systems have coupled equations of motion. However, the iterative method may suffer a low convergence rate, especially when dealing with complex systems with a large number of wheels. To remedy this, [PERSON] and [PERSON] [43] developed a versatile element method called the VBI element, capable of solving systems with multiple interaction forces by formulating them based on the degrees of freedom (DOF) of the wheels. In this approach, the sub-matrices of the beam element bearing wheels are updated at each time step during analysis.