Large Deformation Settlement Series Solution and Numerical Simulation for Underpass Project Track

Abstract

The settlement of track in a railroad underpass project was studied by theoretical analysis and numerical simulation, based on the model of winkle beam on elastic foundation. Taking the settlement of roadbed as a nonlinear boundary condition, the control differential equation of the track large deformation settlement was established and solved by the Fourier series method; the numerical solution was obtained by establishing a numerical model based on an actual project and compared and analyzed with the series solution and cosine function solution. The correlation coefficients of the track settlement curves obtained from the series solution and the numerical solution were higher than those of the cosine function solution while the numerical solution with the maximum track settlement difference was only 0.8% of the maximum settlement of the track numerical solution. The accuracy of the series solution method is higher in this paper, and the track settlement curve obtained from theoretical calculation reflects and predicts the track settlement well when the track settlement cannot be measured directly.

1. Introduction

In recent years, with the rapid development of traffic construction, the traffic safety problems of road and railroad level crossings have become more and more prominent. Therefore, the level crossing of the railroad and road can be changed into an interchange, and the underpass railroad frame culvert (also called box culvert) becomes the main solution to this problem [1,2,3,4]. The process of box culvert jacking or subway tunnel shield construction will inevitably cause disturbance of the surrounding soil, which in turn leads to different degrees of ground displacement and deformation [5,6,7], causing loss of the existing equilibrium, and then triggering the settlement of soil, track or pipeline. However, in practical engineering, for beam type structures such as tracks or pipelines, the settlement cannot be measured directly due to the on-site construction, therefore, the soil settlement under the beam is measured to predict the settlement of the beam in the underpass project. This settlement of the beam becomes a practical and effective method.

The main research methods for the displacement of soil and adjacent track pipeline caused by jacking or shield construction are as follows: (1) the formula method [8,9,10,11]; (2) the numerical analysis method [12,13,14,15,16]. For the settlement analysis of the track, the elastic foundation beam model is usually used. In recent years, many scholars have conducted in-depth research on the calculation theory and method of the foundation beam [17,18,19], among which the Winkler model is widely used by the engineering community because of its simplicity and practicality. The Winkler foundation model assumes that the pressure at a point on the foundation is proportional to the displacement at that point. The frictional resistance between the foundation and the beam is neglected. However, in actual engineering, when the foundation soil is hard and the contact between the beam and the foundation is rough, there is obvious frictional resistance between the two. Second, this effect of frictional resistance between beams and soils is often not directly neglected because the deformation moduli of beams and soils usually have significant differences, which also make the contact mode between them more complicated. He et al. [20] investigated the effect of interfacial friction between the beam and the foundation with numerical examples, and also studied the mechanical behavior due to the tension-free properties of the foundation. The numerical results show that the effects of the tension-free characteristics of the foundation and the interfacial friction have significant effects on the mechanical behavior of the beam-foundation system. Ninik et al. [21] proposed an embedded beam formulation for discrete independent finite element (FE) analysis of pile foundations or rock anchors interacting with the surrounding soil in geotechnical and tunnelling engineering. Xia et al. [22] derived that the lateral deflection of the generalized foundation Timoshenko beam is more sensitive to the compressive stiffness of the Winkler foundation than to the stiffness of the shear layer and rotating springs of the foundation. Horizontal interfacial friction due to foundation adhesion has the same effect as foundation cohesion. It is reasonable to incorporate horizontal interfacial friction into elastic foundations. Craveiro and Neto [23,24] investigated the effects of pipe-soil geometric defect amplitudes and friction on critical loads and post-bending structures. The above research shows that the theoretical solution considering the effect of friction is more accurate, therefore, the effect of the friction effect should be considered in the analysis of the elastic foundation beam.

In addition, the traditional elastic foundation beam model is based on the assumption of small deformation, which does not consider the axial elongation of the beam and the effect of additional axial force generated by the deformation of the beam. However, in the actual underpass construction, the larger foundation settlement also triggers the larger settlement of the upper beam, and the traditional small deformation model is not suitable for the situation where the beam generates large deflection. Thus, it is necessary to establish the control differential equations of the track structure considering the large deformation to make the theoretical calculation more suitable for the actual situation. In terms of large deformation mechanics methods, Yang et al. [25] established the control differential equations considering the axial force of the pipeline, gave the approximate and graded solutions for large deformation of the pipeline, and compared the results of the graded solutions with the approximate solutions to verify the correctness of the graded solutions; however, they did not consider the influence of the frictional effect in the case of large deformation.

Therefore, in this paper, the existing railroad track was regarded as a continuous elastic foundation beam, while the foundation conforms to Winkler’s assumption. The numerical solutions were compared and analyzed. The beam and foundation conformed to Winkler’s assumption; the controlling differential equation was established and the graded solution obtained by considering the large deformation of the foundation beam and the frictional effect between the foundation and the beam, but not considering the debonding of the beam and the foundation. The numerical solution was obtained by numerical simulation based on an actual project, while the graded solution, the numerical solution, and the cosine function solution were compared and analyzed.

2. Theoretical Model of Track Large Deformation Settlement

In the actual underpass project, due to the site construction and safety conditions, the soil settlement under the rail can still be measured directly, but often the track settlement cannot be directly measured [26,27]. So, establishing a theoretical model using the soil settlement under the rail to predict the track settlement is of great significance for the actual project.

2.1. 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:

$$\begin{array}{l}dM+\tau d{x}^{\prime }\cdot \frac{h}{2}-q\cdot d{x}^{\prime }\cdot \frac{dx+d{x}^{\prime }}{2}-\\ Q\cos \frac{d\theta }{2}\cdot \frac{dx+d{x}^{\prime }}{2}-N\cdot \sin \frac{d\theta }{2}\cdot \frac{dx+d{x}^{\prime }}{2}\\ =0\end{array}$$

(1)

where dx is the length of the rail beam before the micro-element is stressed; dx′ 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; dθ is the relative turning angle of the left and right cross sections after micro-element deformation; τ is the frictional force between the beam and the soil.

The relative turning angle dθ 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 ($dx$ + $d{x}^{\prime }$)/2 = $d{x}^{\prime }$. Neglecting the effect of second-order differentiation, Equation (1) can be simplified as

$$dM+\tau d{x}^{\prime }\cdot \frac{h}{2}-Q\cdot d{x}^{\prime }=0$$

(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

$$q=Kb\left(S-w\right)-g\gamma$$

(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; γ 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 horizontal, the size of the drag force τ becomes linearly distributed. Figure 2 shows the distribution of the drag force, then there is the following:

$$\tau \left(x\right)=\frac{\left(L-x\right)}{L}{\tau }_0$$

(4)

$${\tau }_0=fKb{w}_0$$

(5)

where L is half of the calculated length; f is the coefficient of friction; w₀ 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. From the equilibrium of forces in the y-direction, and simplifying it, we obtain

$$Nd\theta -dQ+qd{x}^{\prime }=0$$

(6)

Setting the track ratio u = dx′/dx (the ratio of the length of the track after deformation to the length before deformation), from the relationship between bending moment and deflection the following can be obtained:

$$M=\frac{EI}{u}\frac{d\theta }{dx}=\frac{EI}{u}\frac{d^{2}w}{d{x}^2}$$

(7)

Combining Equation (7) with Equation (2) gives

$$Q=\frac{EI}{u^2}\frac{d^{3}w}{d{x}^3}+\tau \cdot \frac{h}{2}$$

(8)

Substituting Equations (4), (5) and (8) into Equation (6), the following differential equation is obtained:

$$\frac{EI}{u^2}\frac{d^{4}w}{d{x}^4}-N\frac{d^{2}w}{d{x}^2}=uq+\frac{h}{2L}fKb{w}_0$$

(9)

2.2. Boundary Conditions

The track is considered a long beam with both ends (x = ±L) solidly supported and the angle of rotation at the midpoint of the beam (x = 0) is 0. Therefore, the boundary condition of the beam can be written as

$$\frac{dw}{dx}|{}_{x=+L}=\frac{dw}{dx}|{}_{x=-L}=\frac{dw}{dx}|{}_{x=0}=0$$

(10)

The vertical displacement of the soil layer can be calculated by Peck’s formula, which is as follows:

$$S\left(x\right)=-S_0{e}^{-\frac{x^2}{2i^2}}$$

(11)

where S₀ is the maximum settlement of the soil layer; i is the horizontal distance from the inverse bend point of the soil settlement curve to the midpoint.

2.3. Fourier Series Solution

Assuming that the settlement curve of the track can still be calculated using Peck’s formula but only its track ratio and axial force, i.e.,

$$w\left(x\right)=-w_0{e}^{\frac{x^2}{2j^2}}$$

(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′)² and omitting higher order terms for its McLaughlin expansion gives the following:

$${\displaystyle {\int }_{l}ds={\displaystyle {\int }_{-L}^L\sqrt{1+{\left(w^{\prime }\right)}^2}}}dx\approx {\displaystyle {\int }_{-L}^L\left(1+\frac{1}{2}t\right)dx}$$

(13)

$t=\frac{w_0^2}{j^4}{x}^2{e}^{\frac{-x^2}{j^2}}$. Substituting this into Equation (13) yields:

$$\begin{array}{l}{\displaystyle {\int }_{-L}^L\left(1+\frac{1}{2}\frac{w_0^2}{j^4}{x}^2{e}^{\frac{-x^2}{j^2}}\right)dx\approx 2L+\frac{w_0^2}{j^4}}{\displaystyle {\int }_0^{+\infty }{x}^2}{e}^{\frac{-x^2}{j^2}}dx\\ =2L+\frac{w_0^2\sqrt{\pi }}{4j}\end{array}$$

(14)

The calculated length is 2L, so the ratio u is as follows:

$$u=1+\frac{w_0^2\sqrt{\pi }}{8jL}$$

(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

$$N=EA\left(1-\frac{1}{u}\right)$$

(16)

where E is the modulus of elasticity of the track, Pa; A is the cross-sectional area of the track, m².

Substituting Equation (15) into Equation (16) yields

$$N=\frac{w_0^2\sqrt{\pi }}{8jL+w_0^2\sqrt{\pi }}AE$$

(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.,

$$S\left(x\right)=\frac{b_0}{2}+{\displaystyle \sum _{k=1}^{\infty }{b}_k}\cos \frac{k\pi x}{L}$$

(18)

where b₀ 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:

$$\begin{array}{l}{b}_0=\frac{1}{L}{\displaystyle {\int }_{-L}^{L}S\left(x\right)}dx\\ \approx \frac{1}{L}{\displaystyle {\int }_{-\infty }^{\infty }S\left(x\right)}dx=-\sqrt{2\pi }\frac{i}{L}{S}_0\end{array}$$

(19)

$$\begin{array}{l}{b}_k=\frac{1}{L}{\displaystyle {\int }_{-L}^{L}S\left(x\right)\cos \frac{k\pi x}{L}}dx\\ \approx \frac{1}{L}{\displaystyle {\int }_{-\infty }^{\infty }S\left(x\right)}\cos \frac{k\pi x}{L}dx\\ =-\sqrt{2\pi }\frac{i}{L}{S}_0{e}^{-\frac{{\left(k\pi i\right)}^2}{2L^2}}\end{array}$$

(20)

Spread the orbital sedimentation w on [−L, +L] into a cosine series, i.e.,

$$w\left(x\right)=\frac{a_0}{2}+{\displaystyle \sum _{k=1}^{\infty }{a}_k\cos \frac{k\pi x}{L}}$$

(21)

where w(x) is the orbital sedimentation; a₀ 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

$$w^{″}\left(x\right)=-{\displaystyle \sum _{k=1}^{\infty }{a}_k}{\left(\frac{k\pi }{L}\right)}^2\cos \frac{k\pi x}{L}$$

(22)

$${w^{″}}^{″}\left(x\right)={\displaystyle \sum _{k=1}^{\infty }{a}_k}{\left(\frac{k\pi }{L}\right)}^4\cos \frac{k\pi x}{L}$$

(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:

$$a_0=b_0+\frac{hf}{uL}{w}_0-\frac{2}{Kb}g\gamma$$

(24)

$$a_k=\frac{uKb\cdot b_k}{uKb+N{\left(\frac{k\pi }{L}\right)}^2+\frac{EI}{u^2}{\left(\frac{k\pi }{L}\right)}^4}$$

(25)

Note: Equation (21) is the object of the solution in this paper. a₀ 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; w₀ and j in Equation (12) can only be determined by the approximate method at present.

From Equation (21), we can obtain the following:

$$w_0=-\frac{a_0}{2}-{\displaystyle \sum _{k=1}^{\infty }{a}_k}$$

(26)

For the track sink width j, $w^{″}\left(x\right)=0$ it can be obtained from x = j, i.e.,

$${\displaystyle \sum _{k=1}^n{a}_k}{\left(\frac{k\pi }{L}\right)}^2\cos \frac{k\pi j}{L}=0$$

(27)

In order to solve the orbital settlement equation, this can be calculated by the iterative method; the first iteration takes u = 1, N = 0, w₀⁽⁰⁾ = S₀, j⁽⁰⁾ = i, solving the system of equations to get a₀ and a_k, and thus w₀⁽¹⁾ and j⁽¹⁾; the second iteration uses w₀⁽¹⁾ and j⁽¹⁾ to obtain the new u and N, and thus w₀⁽²⁾ and j⁽²⁾; continue until the difference between w₀⁽ⁿ⁾ and j⁽ⁿ⁾ and w₀⁽ⁿ⁻¹⁾ and j⁽ⁿ⁻¹⁾ meets the accuracy requirement.

3. Field Monitoring Data of Track Settlement Deformation

3.1. Project Overview

This paper relies on the Ningbo Hengshan Road leveling project, corresponding to the line center mileage K182 + 218.65, line reinforcement construction range K182 + 180~K182 + 250, 70 m long, hole span arrangement for 4 m + 12 m + 12 m + 4 m, frame culvert under the railroad 7 is shared, and the existing railroad is orthogonal, track number 1 to 7, the top direction from 1 to 7. Figure 3 shows the construction plan layout diagram. The net height of the 4 m frame in the frame body is 6.85 m, and the top plate, bottom plate, and sidewall are 0.45 m thick; the net height of the 12 m frame is 6.0 m, the top plate, and side wall is 0.75 m thick, the bottom plate is 1.0 m thick; the length of a frame is 41.5 m. The size of the frame culvert is shown in Figure 4 and the size unit in the figure is cm.

The soil layer is divided into five uniform layers to the bottom. From top to bottom the soil layers are powder clay, silt, silty clay, powder clay, and clay,

Table 1. Soil parameters.

Name of Soil Layer	Density/kg·m−3	Modulus of Elasticity/MPa	Poisson’s Ratio	Friction Angle/°	Cohesion/kPa	Thickness/m
Powdery clay	1800	15	0.3	25	10	3
Silt	1300	2	0.3	6	8	2
Silty clay	1600	10	0.3	15	25	10
Powdery clay	1900	18	0.3	25	10	10
Clay	2000	30	0.35	15	25	25

shows the soil layer parameter table.

3.2. Monitoring Point Layout

In order to effectively predict and analyze the settlement deformation of the track and roadbed during the construction process and ensure the safe and smooth completion of the box culvert jacking construction, site monitoring becomes an indispensable part of the project. Considering the actual situation on site, the monitoring points are arranged as follows: four monitoring points are arranged equidistantly on the temporary beam; one monitoring point is arranged on the temporary pier; one monitoring point is set at each end of the strip foundation, and one monitoring point is set in the middle of each of the two tracks in the middle of the strip foundation. The plan layout of the monitoring points is roughly displayed in Figure 5.

4. Numerical Calculation of Track Settlement Deformation Based on ABAQUS

ABAQUS software was used to establish a 3D solid model with a scale of 1:1 for finite element calculation to analyze the effect of underpass jacking of the frame culverts on the roadbed and track settlement deformation. Considering the symmetry of the structure, to reduce the calculation volume of the model, the model was selected to calculate half of the structure. The model mainly contains a frame culvert, roadbed soil layer, and track structure, among which the track structure consists of three components: track, rail sleeper, and roadbed. The size and material properties of each component of the model are set and selected according to the construction design drawings and geological survey data.

Considering the boundary effect, the plane size of the overall model is 80 × 44 m, the height is 50 m from the roadbed surface to the bottom of the soil layer [4,28], the top surface of the model is free, the surrounding boundary is a horizontal constraint, the bottom surface is a three-way constraint, the symmetry surface is a symmetry constraint. The frame culvert structure is simulated by solid units, and the bed adopts wedge-shaped units, with an approximate global size of 4 and several individual component units of 120; the track, rail sleeper, frame, and roadbed soil layer all adopt hexahedral units, with the approximate global size of 1, 0.5, 2, and 2, with the number of individual component units of 1440, 12, 2329 and 20,160 respectively. A schematic diagram of the model is shown in Figure 6.

Since the actual engineering has considerable complexity, it is impossible to maintain complete consistency with the actual engineering when establishing the numerical model. Therefore, according to the focus of the study, the model was simplified under the premise of ensuring the accuracy of the calculation results and reducing the difficulty of the model calculation. The basic assumptions are as follows:

(1)

The layers are simplified to ideal, isotropic, uniform elastoplastic soils with a uniform horizontal distribution of each layer.

(2)

The frictional resistance in the model is uniformly distributed.

(3)

Only the self-weight stress is considered when calculating the initial stress of the soil body, and the structural stress of the soil body is not considered.

(4)

The effect of groundwater level change is not considered due to the deep water table.

(5)

The connection between the rail and the rail sleeper is equivalent to the spring connection.

(6)

The load of rolling stock is ignored.

The process of frame culvert with soil jacking is mainly reflected in the excavation of soil and the contact analysis of the box culvert and soil in the numerical simulation. The friction coefficient is taken according to the site survey report, and the normal contact is “hard contact” [15]. Considering the convergence of the calculation, the model adopts activation of the box culvert at the corresponding position to simulate the jacking process, while the soil in contact with the outer surface of the box culvert applies a surface load to simulate the friction force generated by the box culvert jacking on the soil. The value of the friction coefficient, friction force calculation reference [29,30], the friction coefficient in the finite element calculation are taken as 0.3, 0.5, and 0.7 at the top, side plate, and bottom plate of the box culvert, respectively, and the three normal forces on the soil body are the self-weight of the overlying soil body, the active earth pressure borne by both sides, and the self-weight of the overlying soil body plus the self-weight of the box culvert, respectively. The friction force acting on the soil at the outer wall of the box culvert is 9.68 kPa at the top plate, 24.75 kPa at both sides of the plate, and 46.52 kPa at the bottom plate, which is applied to the soil at the outer wall of the box culvert according to the uniform load, while the friction direction is the same as the top-in direction. After the simulation is completed, the corresponding settlement displacement values of the roadbed and track can be extracted by creating paths. Since the top course of the box culvert is long, seven different working conditions are selected as shown in

Table 2. Working condition formulation table.

Work Condition Serial Number	Characteristics of Working Conditions	Remarks
1	Box culvert crossing the 1st railroad	For each working condition, only the parameters corresponding to that working condition are changed, other parameters remain unchanged
2	Box culvert crossing the 2nd railroad
3	Box culvert crossing the 3rd railroad
4	Box culvert crossing the 4th railroad
5	Box culvert crossing the 5th railroad
6	Box culvert crossing the 6th railroad
7	Box culvert crossing the 7th railroad

.

5. Results and Discussion

5.1. Analysis of Numerical Simulation Results

5.1.1. Analysis of Longitudinal Settlement Pattern of Roadbed and Track

The longitudinal direction of the roadbed and track is indicated perpendicular to the direction of the roadbed or track travel. To explore the settlement law of each roadbed and track section under different working conditions during the jacking process of the box culvert, the maximum settlement values of roadbed and track under each working condition in the model are selected and plotted according to the order of working conditions, as shown in Figure 7.

From Figure 7, we can see that the roadbed and the corresponding track change law are basically the same. The maximum settlement change of roadbed and track happens to occur when the box culvert crosses this roadbed, and the settlement change of the roadbed and track is also larger when the box culvert crosses the adjacent roadbed of this roadbed. For example, the maximum settlement change of roadbed 1 is influenced by working conditions 1, 2, and 3, and the maximum settlement change of roadbed 4 is influenced by working conditions 3, 4, and 5. Due to the unloading rebound after soil excavation, when the box culvert through roadbed 4, causes a small amount of rebound in the maximum settlement value of roadbed and track 1 and 2, the maximum rebound is about 2 mm.

5.1.2. Analysis of Transverse Settlement Pattern of Roadbed and Track

(1) Analysis of transverse settlement pattern of roadbed

Roadbed and track transverse are said to be parallel to the direction of the roadbed or track travel. In order to explore the transverse settlement law of each roadbed cross-section, the model is selected at the end of the working conditions of each roadbed transverse settlement and the values plotted to give the roadbed transverse settlement curve as shown in Figure 8. From Figure 8 it can be seen that after the end of jacking, the final settlement curve of each roadbed cross-section is approximately the same, all are in line with the form of a Gaussian curve; the maximum settlement interval of the roadbed is (−61 mm, −68 mm), the main settlement range interval is (−40 m, 40 m).

(2) Analysis of transverse settlement pattern of track

To investigate the transverse settlement law of each track, the transverse settlement value of each track after the end of the working condition in the model is selected and plotted, while the transverse settlement curve of the track is shown in Figure 9. It can be seen from Figure 9 that after the end of jacking, the final settlement curve of each track transverse section is about the same; all conform to the form of a Gaussian curve; the maximum settlement interval of the track is (−60 mm, −66 mm), and the main settlement range interval is (−40 m, 40 m).

5.2. Comparison Analysis between Measured and Numerical Solutions of Roadbed Settlement

In order to study the track settlement law of the numerical model, the actual measured data of roadbed 4 and roadbed 7 are selected to compare and analyze with the numerical simulation results, while the numerical analysis simulation results are called numerical solutions. Roadbed 4 is located at the midpoint of the entire topping area and is most obviously disturbed by the box culvert topping; the settlement of roadbed 1 is affected by both the excavation of the adjacent pit and the box culvert topping, while the settlement of roadbed 7 is only caused by the box culvert topping, which better reflects the change of roadbed settlement caused by topping. Therefore, the measured data of roadbed 4 and roadbed 7 were selected for comparison.

The measured data of roadbed 4 and roadbed 7 are plotted with the numerical solution of the final settlement obtained at the end of the numerical jacking simulation, as shown in Figure 10 and Figure 11, respectively.

From Figure 10 and Figure 11, it can be seen that the measured data curves of each cross section and the numerical solution curves match and are roughly in line with the Peck curve. The measured settlement is larger than the numerical solution, with the main reasons as follows: Because the numerical simulation is in an ideal state, it cannot fully consider all the factors that can affect the soil during the construction; the location of the box culvert and the upper space of the soil are fully excavated, and the loss rate of the soil is larger than that of the upper cover of the box culvert, resulting in a larger measured settlement value. The main reason that the measured values are not completely symmetrical is that during the actual jacking process, the jacking direction often has a slight deviation and correction, and the box culvert is approximately “snake” forward, which leads to the measured values on both sides not being completely symmetrical.

5.3. Comparison of Measured Value, Numerical Solution, Fourier Series Solution, and Cosine Function Solution

The box culvert jacking track is erected in the upper part of the track, and has a good connection with the track. Because the settlement value of the track cannot be measured directly, the site through the monitoring of the displacement value of the beam achieves the monitoring of the track settlement, erected in the upper part of the track beam distribution on both sides of the track. Therefore, the average value of both sides of the beam is taken as the settlement value of the track.

The work point data of the project is adopted as follows: track linear density is 60 kg/m, cross-sectional area is 77.45 cm², cross-sectional moment of inertia is 3217 cm⁴, modulus of elasticity is 210 GPa, the friction coefficient is 0.3, foundation coefficient is 50 MPa/m, gravitational acceleration is taken as 9.8 m/s², beam height is 0.176 m, track width is 1.435 m, calculated length of Half L = 5i.

The settlement curves of different roadbeds can be obtained by numerical simulation, which is used as the nonlinear boundary condition for the previous level solution, i.e., S₀ is the maximum settlement value of different roadbeds obtained by numerical simulation. The series solution of track settlement can be obtained by using the theoretical model of track settlement to realize the prediction of track settlement through roadbed settlement, where the series solution of track settlement is obtained by iterative calculation using MATLAB programming. Letting the maximum settlement value of the roadbed remain unchanged, a very strongly correlated Peck roadbed settlement curve can be determined by changing the size of the settlement trough, which can be obtained by fitting the following:

$$S\left(x\right)=-68.96\times e^{-\frac{x^2}{2\times {18}^2}}$$

(28)

$$S\left(x\right)=-73.52\times e^{-\frac{x^2}{2\times {14.5}^2}}$$

(29)

For comparative analysis, the cosine function formula of the track surface settlement deformation caused by uneven settlement of the roadbed can be used to calculate the track surface settlement under the condition that the rail sleeper is not hanging as proposed by the literature [31]. The formula is as follows:

$$w(x)=-\frac{\delta }{2}\left(1+\cos \frac{2\pi }{l}x\right)$$

(30)

where δ is the wave amplitude, mm; l is the wavelength, m.

$$l_2=l_1+\frac{2\left(b^{\prime }+h^{\prime }\right)}{\tan \phi }$$

(31)

where l₁ is the wavelength of the full-wave cosine type roadbed uneven settlement, m; l₂ is the wavelength of the full-wave cosine type track settlement slot, m; φ is the diffusion angle of ballast deformation; $b^{\prime }$ is the thickness of ballast, m; $h^{\prime }$ is the height of rail sleeper, m.

$${\delta }_2=\frac{{\delta }_1{l}_1}{l_2}$$

(32)

where δ₁ is the wave amplitude of the full-wave cosine type roadbed uneven settlement, mm; δ₂ is the wave amplitude of the full-wave cosine type track settlement slot, mm.

According to the results of the model test in the literature [31], the deformation diffusion angle of the roadbed settlement transferred to the rail surface after the action of ballast and rail is 25°, i.e., φ = 25°. According to the actual situation of this project, the ballast thickness is 0.35 m, the rail sleeper height is 0.28 m, the wavelength l₁ of uneven settlement of the roadbed is 120 m, the wave amplitude of uneven settlement of the roadbed 4 and roadbed 7 are 68.96 mm and 73.52 mm respectively. Using the Formula (31), it can be found that the wavelength l₂ of rail surface settlement trough is 122.7 m, using (32) the rail 4 and rail 7 can be found The wave amplitudes of the track surface settlement groove are 67.44 mm and 71.9 mm respectively, and the result obtained from literature [31] is called the cosine function solution. Therefore, the cosine function settlement curves of track 4 and track 7 are as follows:

$$w\left(x\right)=-\frac{67.44}{2}\left(1+\cos \frac{2\pi }{122.7}x\right)$$

(33)

$$w\left(x\right)=-\frac{71.9}{2}\left(1+\cos \frac{2\pi }{122.7}x\right)$$

(34)

Note: The solution calculated by the theoretical model of the orbit in this paper is the series solution, the solution obtained by the model test in the literature [31] is the cosine function solution, the settlement of the orbit in the numerical simulation results is the numerical solution. The numerical solution, the series solution, and the cosine function solution are three different results of the solution of the orbit settlement curve.

The measured value, numerical solution, the series solution, and the cosine function solution are plotted as curves as shown in Figure 12 and Figure 13.

From Figure 12 and Figure 13, the following can be seen

The measured value of the settlement is smaller than the numerical solution; the main reason is that the railroad daily maintenance needs to adjust the track to ensure the safety of traffic, each adjustment makes the track restore to the initial horizontal position, that is, by letting the beam settle to zero, means the site monitoring does not monitor all the displacement changes of the beam; the measured value of the beam cannot fully reflect the settlement displacement of the track.

The maximum settlement of the orbit for the numerical solution, the series solution, and the cosine function solution of orbit 4 is 68.21 mm, 68.75 mm, and 67.44 mm respectively; the maximum settlement of the orbit for the numerical solution, the series solution, and the cosine function solution of orbit 7 is 72.49 mm, 73.06 mm, and 71.9 mm respectively. The difference between the maximum settlement of orbital tracks of orbital tracks 4 and 7 is 0.54 mm, 0.77 mm, 0.57 mm, and 0.59 mm, which are about 0.8%, 1.1%, 0.8%, and 0.8% of the maximum settlement of orbital tracks of the numerical solution, so the accuracy of the maximum settlement of orbital tracks is high.

The correlation coefficients of the series solution and cosine function solution of tracks 4 and 7 with the numerical solution are 99.3%, 94.28% and 99.23%, 88.31%, respectively. The main settlement range widths of the orbital settlement curves of the graded and numerical solutions are closer and match better; the differences between the main settlement range widths of the orbital settlement curves of the cosine function solution and the numerical solution are mainly related to the size of the wavelength taken.

6. Conclusions

(1)

Relying on the actual frame culvert jacking project, a 1:1 3D finite element model was established, and the numerical solutions of roadbed and track settlement were obtained. At the same time, the measured values of roadbed settlement were compared, and it was found that the measured data curves of each cross section and the numerical solution curves basically matched each other, which were roughly in line with the Peck curve.

(2)

The correlation coefficients (99.3% and 99.23%) of the track settlement curves obtained from the numerical solution and the numerical solution are higher and in better agreement than those obtained from the cosine function solution, and the maximum track settlement difference is only 0.8% of the maximum track settlement from the numerical solution. The theoretical method in this paper is more accurate, and the calculated track settlement curve can predict and reflect the actual settlement of the track.