# Large Deformation Settlement Series Solution and Numerical Simulation for Underpass Project Track
## 2 Theoretical Model of Track Large Deformation Settlement
### Fourier Series Solution  
Assuming that the settlement curve of the track can still be calculated using [PERSON]'s formula but only its track ratio and axial force, i.e.,  
\[w(x)=-w_{0}e^{\frac{x^{2}}{2^{\beta/2}}} \tag{12}\]  
Based on assumption (4), take L to be more than 3 times the sink \(i\). Curve integration of Equation (12) over \([-L,+L]\) such that \(t=(w^{\prime})^{2}\) and omitting higher order terms for its McLaughlin expansion gives the following:  
\[\int_{l}ds=\int_{-L}^{L}\sqrt{1+(w^{\prime})^{2}}dx\approx\int_{-L}^{L}\left(1 +\frac{1}{2}t\right)dx \tag{13}\]  
\(t=\frac{w_{0}^{2}}{\beta^{2}}x^{2}e^{\frac{-x^{2}}{\beta^{2}}}\). Substituting this into Equation (13) yields:  
\[ \begin{split}&\int_{-L}^{L}\left(1+\frac{1}{2}\frac{w_{0}^{2}}{ \beta^{2}}x^{2}e^{\frac{-x^{2}}{\beta^{2}}}\right)dx\approx 2L+\frac{w_{0}^{2}}{ \beta^{2}}\int_{0}^{+\infty}x^{2}e^{\frac{-x^{2}}{\beta^{2}}}dx\\ &=2L+\frac{w_{0}^{2}\sqrt{\pi}}{4\beta}\end{split}  \tag{14}\]  
The calculated length is \(2L\), so the ratio \(u\) is as follows:  
\[u=1+\frac{w_{0}^{2}\sqrt{\pi}}{8 jL} \tag{15}\]  
where \(j\) is the sink width of the track, that is, the horizontal distance of the sink curve reverses the bend point from \(x=0\). The relationship between axial force and orbital ratio is given by  
\[N=EA\left(1-\frac{1}{u}\right) \tag{16}\]  
where \(E\) is the modulus of elasticity of the track, Pa; \(A\) is the cross-sectional area of the track, m\({}^{2}\). Substituting Equation (15) into Equation (16) yields  
\[N=\frac{w_{0}^{2}\sqrt{\pi}}{8 jL+w_{0}^{2}\sqrt{\pi}}\,AE \tag{17}\]  
Since the soil settlement and the beam end restraint are symmetrical about the center of the settlement tank, the track settlement is also bound to be symmetrical. In order to solve the Fourier series of Equation (9), it is necessary to expand the \(S\) and \(w\) in Equation (9) into a triangular series. The soil settlement \(S\) is expanded into a cosine series on \([-L,+L]\), i.e.,  
\[S(x)=\frac{b_{0}}{2}+\sum_{k=1}^{\infty}b_{k}\cos\frac{k\pi x}{L} \tag{18}\]  
where \(b_{0}\) and \(b_{k}\) are the coefficients of the Fourier series; \(k\) is a natural number. To make the calculation range large enough, \(L\) is generally taken to be \(3i\) or more. Using Equation (11) we have [16] the following:  
\[ \begin{split}& b_{0}=\frac{1}{\beta}\int_{-L}^{L}S(x)dx\\ &\approx\frac{1}{\beta}\int_{-\infty}^{\infty}S(x)dx=-\sqrt{2\pi} \,i\,S_{0}\end{split}  \tag{19}\]\[ \begin{array}{l}b_{k}=\frac{1}{L}\int_{-L}^{L}S(x)\cos\frac{k\pi x}{L}dx\\ \approx\frac{1}{L}\int_{-\infty}^{\infty}S(x)\cos\frac{k\pi x}{L}dx\\ =-\sqrt{2\pi}i\frac{\ }{L}S_{0}e^{-\frac{(k\pi)^{2}}{2L^{2}}}\end{array}  \tag{20}\]  
Spread the orbital sedimentation w on \([-L,+L]\) into a cosine series, i.e.,  
\[w(x)=\frac{a_{0}}{2}+\sum_{k=1}^{\infty}a_{k}\cos\frac{k\pi x}{L} \tag{21}\]  
where \(w(x)\) is the orbital sedimentation; \(a_{0}\) and \(a_{k}\) are the coefficients of the Fourier series to be determined. From Equation (21), the derivatives of each order of the orbital subsidence can be found as  
\[w^{\prime\prime}(x)=-\sum_{k=1}^{\infty}a_{k}\bigg{(}\frac{k\pi}{L}\bigg{)}^{ 2}\cos\frac{k\pi x}{L} \tag{22}\]  
\[w^{\prime\prime\prime}(x)=\sum_{k=1}^{\infty}a_{k}\bigg{(}\frac{k\pi}{L} \bigg{)}^{4}\cos\frac{k\pi x}{L} \tag{23}\]  
Substitute Equations (19) and (20) into Equation (18), then substitute Equations (3), (18), (21)-(23) into Equation (9), and combine like terms to simplify to obtain the following:  
\[a_{0}=b_{0}+\frac{hf}{uL}w_{0}-\frac{2}{Kb}g\gamma \tag{24}\]  
\[a_{k}=\frac{uKb\cdot b_{k}}{uKb+N\bigg{(}\frac{k\pi}{L}\bigg{)}^{2}+\frac{EI}{ u^{2}}\bigg{(}\frac{k\pi}{L}\bigg{)}^{4}} \tag{25}\]  
Note: Equation (21) is the object of the solution in this paper. \(a_{0}\) and \(a_{k}\) in Equation (21) are determined by the level method; Equation (12) is only used to approximate the axial force and orbital ratio; \(w_{0}\) and \(j\) in Equation (12) can only be determined by the approximate method at present.