# An Effective Tangent Stiffness of Train-Track-Bridge Systems Based on Artificial Neural Network
## 2 The WRI Element and the BP Neural Network  
### The Internal Resisting Force and Consistent Tangent Stiffness of the WRI Element  
The WRI element consists of a wheel node and all the rail nodes (\(b_{1}\), \(b_{2}\)... \(b_{i}\)... ) that the wheel node may pass through, and the crucial part of the WRI element [8] is the so-called active portion, as can be seen in Figure 1.  
The active portion contains one wheel node and two rail nodes, marked as \(O^{w}\), \(b_{i}\), and \(b_{i+1}\), respectively. Then, by projecting \(O^{w}\) vertically onto the rail element, a so-called  
Figure 1: The WRI element and its active portion. 'virtual' rail node \(O^{r}\) can be obtained. The coordinates of \(O^{r}\) can be obtained by linear interpolation from the coordinates of \(b_{i}\) and \(b_{i+1}\), which can be expressed as  
\[\left\{ \begin{array}{l}X_{1}^{O^{r}}=N_{1}^{t}X_{1}^{b_{1}}+N_{2}^{t}X_{1}^{b_{ 1}+1}\\ X_{2}^{O^{r}}=N_{1}^{t}X_{2}^{b_{2}}+N_{1}^{t}X_{2}^{b_{i}}+N_{3}^{t}X_{2}^{b_{ i+1}}+N_{4}^{t}X_{6}^{b_{i+1}}\\ X_{3}^{O^{r}}=N_{1}^{t}X_{3}^{b_{1}}+N_{2}^{t}X_{5}^{b_{1}}+N_{3}^{t}X_{3}^{b_{ i+1}}+N_{4}^{t}X_{5}^{b_{i+1}}\\ X_{4}^{O^{r}}=N_{1}^{t}X_{4}^{b_{i}}+N_{1}^{t}X_{4}^{b_{i+1}}\\ X_{5}^{O^{r}}=N_{1}^{t}X_{3}^{b_{1}}+N_{2}^{t}X_{3}^{b_{i}}+N_{3,1}^{t}X_{3}^{b _{i+1}}+N_{4,1}^{t}X_{5}^{b_{i+1}}\\ X_{6}^{O^{r}}=N_{1,1}^{t}X_{2}^{b_{i}}+N_{2,1}^{t}X_{6}^{b_{i}}+N_{3,1}^{t}X_{2 }^{b_{i+1}}+N_{4,1}^{t}X_{6}^{b_{i+1}}\end{array} \right. \tag{1}\]  
where \(X_{j}^{i}\) denotes the \(j\)th coordinate component of the point \(i\), \(N_{I}^{L}\) (\(I=1\), \(2\)) denotes the shape function of the linear interpolation, \(N_{I}^{H}\) (\(I=1\), \(2\), \(3\), \(4\)) denotes the shape function of the Hermite interpolation, and \(N_{1,1}^{H}\) (\(I=1\), \(2\), \(3\), \(4\)) denotes the derivative of the Hermite interpolation shape functions with respect to \(x\). In each time step, the wheel node moves forward, and the shape functions change. After the wheel node \(O^{w}\) crosses \(b_{i+1}\), the active position is updated immediately and consists of \(O^{w}\), \(b_{i+1}\), and \(b_{i+2}\). The element internal resisting force \(\mathbf{R}\) consists of the forces on the nodes \(O^{w}\), \(b_{i}\), and \(b_{i+1}\), i.e., \(\mathbf{R}\)\(=[\mathbf{R}_{w},\mathbf{R}_{b_{i}},\mathbf{R}_{b_{i+1}}]\) and can be obtained based on the nodal displacement \(\mathbf{u}\) and velocity \(\mathbf{v}\), i.e., \(\mathbf{R}=\mathbf{R}(\mathbf{u},\mathbf{v})\), where the nodal displacement \(\mathbf{u}\) consists of the displacements of the wheel node and two rail nodes, i.e., \(\mathbf{u}\)\(=[\mathbf{u}_{w},\mathbf{u}_{b_{i}},\mathbf{u}_{b_{i+1}}]\), while the velocity \(\mathbf{v}\) is a function of \(\mathbf{u}\), i.e., \(\mathbf{v}=\mathbf{v}(\mathbf{u})\). It is worth mentioning that the function can be explicitly obtained based on the approximation of velocity using displacement when a time-stepping discretization (e.g., Newmark method) is employed. The element internal resisting force \(\mathbf{R}\) can be calculated based on the contact force \(\mathbf{F}\) between wheel node \(O^{w}\) and virtual rail node \(O^{r}\), which consists of the normal contact force \(F_{n}\) and the tangential contact force \(F_{t}\), i.e.,  
\[\mathbf{R}=f(\mathbf{F})=f(F_{n}(X(\mathbf{u})),F_{t}(\mathbf{u},\mathbf{v})) \tag{2}\]  
where \(X\) denotes the coordinates obtained by element nodal displacement \(\mathbf{u}\). Suppose there is a virtual interpenetration between the rail and wheel profile, as can be found in Figure 2. The normal contact force \(F_{n}^{c}\) and tangential contact forces \(F_{\Pi}^{c}\) and \(F_{t2}^{c}\) are calculated using nonlinear Hertz theory, [PERSON] linear theory, and the Polach formulation [19]. After the contact force \(\mathbf{F}\) is obtained, the resisting force on the wheel node, \(\mathbf{R}_{w}\) and the resisting force on the virtual rail node \(\mathbf{R}_{r}\) (in Figure 3) can be calculated respectively by  
\[