# Large Deformation Settlement Series Solution and Numerical Simulation for Underpass Project Track
## 2 Theoretical Model of Track Large Deformation Settlement
### Establishment of Differential Equations  
To establish the theoretical model for predicting track settlement, the track is assumed to be a continuous elastic foundation beam, the stiffness of the components under the track is integrated as the foundation coefficient, the foundation conforms to the Winkler assumption, and the unit track micro-element is taken for the force analysis, as shown in Figure 1.  
Taking the moment at the center of the right-hand section of the micro-element gives the following:  
\[ \begin{array}{l}dM+\tau dx^{\prime}\cdot\frac{h}{2}-q\cdot dx^{\prime}\cdot \frac{dx+dx^{\prime}}{2}-\\ Q\cos\frac{d\theta}{2}\cdot\frac{dx+dx^{\prime}}{2}-N\cdot\sin\frac{d\theta}{2} \cdot\frac{dx+dx^{\prime}}{2}\\ =0\end{array}  \tag{1}\]  
where \(dx\) is the length of the rail beam before the micro-element is stressed; \(dx^{\prime}\) is the length after loading; \(h\) is the height of the beam; \(N\) is the cross-sectional axial force; \(Q\) is the section shear force; \(M\) is the section bending moment; \(q\) is the external load; \(d\theta\) is the relative turning angle of the left and right cross sections after micro-element deformation; \(\tau\) is the frictional force between the beam and the soil. The relative turning angle \(d\theta\) of the cross section after the deformation of the rail beam micro-element is small and can be regarded as an infinitesimal amount. The rail beam micro-element deformation is small and can be taken as \((dx+dx^{\prime})/2=dx^{\prime}\). Neglecting the effect of second-order differentiation, Equation (1) can be simplified as  
\[dM+\tau dx^{\prime}\cdot\frac{h}{2}-Q\cdot dx^{\prime}=0 \tag{2}\]  
Unlike the linear control differential equation established by the small deformation theory, the control differential equation considering large deformation and frictional resistance is highly nonlinear. To reduce the degree of nonlinearity of the equation so that it can be solved by the series method, the following reasonable assumptions were made. (1) The track is regarded as a continuous elastic foundation beam, without considering the occurrence of debonding between the track and the lower soil layer, and the force of the soil layer on the track is perpendicular to the track, without the component parallel to the track. The load \(q\) can be calculated according to the Winkler foundation reaction force as  
\[q=Kb(S-w)-g\gamma \tag{3}\]  
where \(K\) is the foundation factor; \(b\) is the track width; \(S\) is the settlement of soil layer; \(w\) is the track settlement; \(g\) is the acceleration of gravity; \(\gamma\) is the orbital line density. (2) By ignoring the influence of the deformation of the bottom of the track on the direction of the drag force, that is, the direction of the drag force is always considered to be  
Figure 1: Schematic diagram of track micro-element stress. horizontal, the size of the drag force \(\tau\) becomes linearly distributed. Figure 2 shows the distribution of the drag force, then there is the following:  
\[\tau(x)=\frac{(L-x)}{L}\tau_{0} \tag{4}\]  
\[\tau_{0}=fKbw_{0} \tag{5}\]  
where \(L\) is half of the calculated length; \(f\) is the coefficient of friction; \(w_{0}\) is the vertical displacement of the track at \(x\) = 0; \(x\) denotes the distance from the position to the midpoint (point \(O\)), m.  
(3) The track deformation satisfies the assumption of a flat section, ignoring the change of section size after the track deformation, that is, the track bending and tensile stiffness remains unchanged. (4) The axial deformation and axial force of the track within a certain range near the sink is constant, and the axial deformation and axial force at the far end can be neglected.