# A Vehicle-Bridge Interaction Element: Implementation in ABAQUS and Verification
## 2 Formulations for Vehicle-Bridge Interaction (VBI) Element Applied in ABAQUS
### Implementation in ABAQUS through UEL Subroutine  
To implement the VBI element theory in ABAQUS, we propose a user-defined element (UEL) subroutine, in which 2-node beam elements are applied. The Hermite interpolation functions are based on the shape functions of the beam element:  
\[N_{1}=1-3\bigg{(}\frac{x}{l_{e}}\bigg{)}^{2}+2\bigg{(}\frac{x}{l_{e}}\bigg{)}^{3} \tag{9}\]  
\[N_{2}=l_{e}\bigg{[}-\bigg{(}\frac{x}{l_{e}}\bigg{)}+2\bigg{(}\frac{x}{l_{e}} \bigg{)}^{2}-\bigg{(}\frac{x}{l_{e}}\bigg{)}^{3}\bigg{]} \tag{10}\]  
\[N_{3}=3\bigg{(}\frac{x}{l_{e}}\bigg{)}^{2}-2\bigg{(}\frac{x}{l_{e}}\bigg{)}^{3} \tag{11}\]  
\[N_{4}=l_{e}\bigg{[}\bigg{(}\frac{x}{l_{e}}\bigg{)}^{2}-\bigg{(}\frac{x}{l_{e}} \bigg{)}^{3}\bigg{]} \tag{12}\]  
where \(x\) is the element local coordinate, and \(l_{e}\) represents element length. It should be noticed that the shape functions applied in this work are based on the Euler-Bernoulli beam theory, assuming that the cross-sections remain perpendicular to the beam axis during deformation, implying that the shear strain is negligible. As a result, the Euler-Bernoulli beam theory tends to provide accurate results for beams with low slenderness ratios and negligible shear effects. For beams with high slenderness ratios or subjected to significant shear forces, it becomes essential to consider higher-order theories, such as the Timoshenko beam theory or shear deformable beam theories, which account for shear deformation effects more accurately. The parameters of the Newmark time-stepping method are also defined in [43]. When dealing with the VBI element, we apply a loop to judge which wheel is on the current element. The element will become a VBI element as long as the position of one wheel is within the region of the coordinates:  
\[ \begin{array}{l}x_{n}=x_{0}(i)\\ x_{c}=x_{n}+vt\end{array}  \tag{13}\]  
where \(x_{n}\) represents the initial position of the \(i\)th wheel, \(t\) denotes time, and \(x_{c}\) represents the current position of the \(i\)th wheel. Based on [PERSON]'s theory [43], we update stiffness and mass matrices through Equations (4)-(8) by moving matrix terms from right to left:  
\[[M]\{\bar{D}\}_{t+\Delta t}+[C]\{D\}_{t+\Delta t}+[K]\{D\}_{t+\Delta t}=\{F_{ b}\}_{t+\Delta t}-\{P_{c}^{*}\}_{t+\Delta t}-\{Q_{c}^{*}\} \tag{14}\]  
where  
\[ \begin{array}{l}[M]=[M_{\rm b}]+[M_{\rm c}^{*}]=\sum[m_{\rm bj}]+\sum\sum \Big{[}m_{\rm cij}^{*}\Big{]}\\ [C]=[C_{\rm b}]+[C_{\rm c}^{*}]=\sum[c_{\rm bj}]+\sum\sum\Big{[}c_{\rm cij}^{*} \Big{]}\\ [K]=[K_{\rm b}]+[K_{\rm c}^{*}]=\sum[k_{\rm bj}]+\sum\Big{[}k_{\rm cij}^{*} \Big{]}\\ \{P_{c}^{*}\}_{t+\Delta t}=\sum\{p_{cj}^{*}\}_{t+\Delta t}\\ \{Q_{c}^{*}\}_{t}=\sum\{q_{cj}^{*}\}_{t}\end{array}  \tag{15}\]  
In ABAQUS, the default time-stepping approach--namely, the Hilber-Hughes-Taylor (HHT-\(\alpha\) method) [44] implicit time integration scheme--is utilized in the analysis. In the UEL subroutine, the effective stiffness matrix ([_AMATRX_]) and the residual vector ([_RHS_]) are adopted to compute the increment of DOF as shown below, which is updated in every step:  
\[[\textit{AMATRX}]\{\Delta u\}=\{\textit{RHS}\} \tag{16}\]where [_AMATRX_] can be shown as  
\[[AMATRX]=\frac{1}{\beta\alpha t^{2}}[M]+\frac{(1+\alpha)\gamma}{\beta\alpha t}[C]+( 1+\alpha)[K] \tag{17}\]  
For the residual vector (\(\{RHS\}\)), [PERSON]'s theory is applied for updating the state variables (_SVARS_) and \(\{G\}\) vector:  
\[\{RHS\}=-\left[M\right]\{\ddot{u}\}_{t+\Delta t}+(1+\alpha)\{G\}_{t+\Delta t} \quad-\alpha\{G\}_{t} \tag{18}\]  
\[\{G\}_{t+\Delta t}=-\left[K\right]\{u\}_{t+\Delta t}-\left[C\right]\{\dot{u} \}_{t+\Delta t}+\{f_{e}\}_{t+\Delta t}-\{q_{e}\}_{t} \tag{19}\]  
where \(\{f_{e}\}\) represents the external force, usually gravity. The default value of the numerical damping parameter \(\alpha\) in the HHT method is -1/20. The details of the algorithm are included in the codes that are provided as Supplementary Materials.