# A Vehicle-Bridge Interaction Element: Implementation in ABAQUS and Verification
## 2 Formulations for Vehicle-Bridge Interaction (VBI) Element Applied in ABAQUS
### VBI Element Formulation  
Two sets of equations of motion are introduced for the vehicle and bridge subsystems. The vehicle's equation of motion is presented in lumped matrix form, with the degrees of freedom (DOF) separated into two parts: carriage and wheels. The force vector is divided into the external force, acting on both the carriage and wheels, and the contact force, which only imposes on wheels, as shown in Equation (1) [43],  
\[ \begin{split}\left[\begin{array}{cc}[m_{\text{uu}}]&[m_{\text{ uw}}]\\ [m_{\text{wu}}]&[m_{\text{ww}}]\end{array}\right]\left\{\begin{array}{c} \left\{\begin{array}{c}\left\{\check{d}_{\text{u}}\right\}\\ \left\{\check{d}_{\text{w}}\right\}\end{array}\right\}\\ \left\{\begin{array}{c}\left[c\check{c}_{\text{uu}}\right]&\left[c_{\text{ ww}}\right]\end{array}\right\}\end{array}\right]\left\{\begin{array}{c} \left\{\begin{array}{c}\left\{\check{d}_{\text{u}}\right\}\\ \left\{\check{d}_{\text{w}}\right\}\end{array}\right\}\\ \left\{\begin{array}{c}\left\{\check{d}_{\text{u}}\right\}\end{array}\right\} \end{array}\right\}_{t+\Delta t}\\ +\left[\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\left[k_{\text{uu} }\right]&\left[k_{\text{ww}}\right]\\ \left[k_{\text{ww}}\right]&\left[k_{\text{ww}}\right]\end{array}\right] \left\{\begin{array}{c}\left\{\begin{array}{c}\left\{\check{d}_{\text{u}} \right\}\\ \left\{\check{d}_{\text{w}}\right\}\end{array}\right\}\\ \left\{\begin{array}{c}\left\{\check{d}_{\text{u}}\right\}\end{array}\right\} \end{array}\right\}_{t+\Delta t}\end{split}  \tag{1}\]where \(\{d_{\rm u}\}\) and \(\{d_{\rm w}\}\) denote the DOF of carriage and wheels, respectively; \(\{f_{\rm{fue}}\}\) and \(\{f_{\rm{we}}\}\) represent the external forces acting on the carriage and wheels; and \(\{f_{\rm{c}}\}\) is the contact force vector that is distributed to DOF of carriage and wheels through matrices \([l_{\rm u}]\) and \([l_{\rm w}]\). The interaction force is given in terms of DOF for wheels. Derailment is neglected in the following analysis for simplicity. The displacement, velocity, and acceleration at the time step \(t+\Delta t\) are obtained using [PERSON]'s time-stepping scheme. These quantities are used to express the contact force between the wheels and the track in terms of the wheels' degrees of freedom (DOF). The governing equation of the carriage is expanded to achieve this. As this paper does not consider derailment, the DOF of the wheels is substituted with the DOF of the contact points between the wheels and the rail. According to beam element theory in finite-element analysis [47], the degrees of freedom (DOF) for the contact points can be expressed in terms of the DOF for nodes of the beam element using shape function interpolation, as illustrated in Equations (9)-(12). At the time step \(t+\Delta t\), we suppose there are \(n\) wheels on the rail. In this case, the beam element subjected to wheel load will be transferred to the VBI element. The \(i\)th VBI element equation of motion is provided by:  
\[\left[m_{\rm b}\right]\left\{\dot{d}_{\rm{bi}}\right\}_{t+\Delta t}+\left[c_{ \rm bi}\right]\left\{d_{\rm{bi}}\right\}_{t+\Delta t}+\left[k_{\rm bi}\right] \left\{d_{\rm{bi}}\right\}_{t+\Delta t}\quad=\left\{f_{\rm bi}\right\}_{t+ \Delta t}-\left\{f_{\rm{bei}}\right\}_{t+\Delta t} \tag{2}\]  
where \([m_{\rm bi}]\), \([c_{\rm bi}]\) and \([k_{\rm bi}]\) represent the mass, damping, and stiffness matrices of the ith element of the rail, respectively; \(\{d_{\rm bi}\}\) is the nodal displacement vector; \(\{f_{\rm bi}\}\) is the external force vector; and \(\{f_{\rm{bei}}\}\) is the vector of equivalent nodal forces related to the ith factor of the contact force vector \(\{f_{\rm{c}}\}\):  
\[\left\{f_{\rm{bei}}\right\}_{t+\Delta t}=\{N_{\rm ci}^{\rm v}\}f_{ci,t+\Delta t} \tag{3}\]  
where \(N_{i}^{\rm v}\) denotes the shape function of the ith bridge element, which is represented by Hermitian cubic polynomials.