Theoretical Study on Pipeline Settlement Induced by Excavation of Ultra-Shallow Buried Pilot Tunnels Based on Stochastic Media and Elastic Foundation Beams

Abstract

Excavation of ultra-shallow pilot tunnels triggers surface settlement and endangers surrounding pipelines. The discontinuous settlement curve from traditional stochastic medium theory cannot be directly integrated into the foundation beam model, limiting pipeline deformation prediction accuracy. The key novelty of this study lies in proposing an improved coupled method tailored to ultra-shallow burial conditions: converting the discontinuous settlement solution into a continuous analytical one via polynomial fitting, embedding it into the Winkler elastic foundation beam model, and realizing pipeline settlement prediction by solving the deflection curve differential equation with the initial parameter method and boundary conditions. Four core factors affecting pipeline deformation are identified, with pilot tunnel size as the key. Shallower depth (especially 5.5 m) intensifies stratum disturbance; pipeline parameters (diameter, wall thickness, elastic modulus) significantly impact bending moment, while stratum elastic modulus has little effect on settlement. Verified by the Xueyuannanlu Station project of Beijing Rail Transit Line 13, theoretical and measured settlement trends are highly consistent, with core indicators meeting safety requirements (max theoretical/measured settlement: −10.9 mm/−8.6 mm < 30 mm; max rotation angle: −0.066° < 0.340°). Errors (max 5.1 mm) concentrate at the pipeline edge, and conservative theoretical values satisfy engineering safety evaluation demands.

1. Introduction

As urban underground space development moves towards the ultra-shallow burial orientation, underground excavation of pilot tunnels poses critical challenges to the safe operation of adjacent underground pipelines. Although the shallow tunneling method can reduce ground traffic interference, under ultra-shallow burial conditions, the stratum stress arch effect is significantly weakened, which easily causes asymmetric stratum deformation, thereby leading to complex settlements (even heave) of adjacent underground pipelines and posing a critical threat to pipeline safety. Therefore, it is necessary to improve the computation theory of pipeline settlement under ultra-shallow burial conditions. Notably, soil cover (or burial depth) is a universally critical factor affecting the interaction between underground structures and strata. Maleska and Beben [1] confirmed that soil cover depth significantly influences the response of underground structures through their study on corrugated steel plate (CSP) bridges, which supports the focus on pilot tunnel burial depth as a core factor in this research.

The Winkler foundation beam model, as a Classic method for parsing pipeline-structure interaction [2], simulates the stratum as a series of standalone springs to characterize soil-pipe interaction. However, its accuracy is critically dependent on the accuracy of the input ground surface settlement curve. In general deep and shallow burial cases, the Peck formula and its improved formulas are often consumed. Based on the Winkler foundation beam theory, Yu Jian et al. [3] proposed a theoretical formula for foundation modulus considering the buried depth of elastic foundation beams through a two-stage analysis method, overcoming the limitations of traditional elastic foundation beam theory that requires assuming the surface of an elastic half-space and vertical forces must act on the center. Based on the modified Winkler foundation model considering the influence of pipeline burial depth, Wang Haitao et al. [4] conducted Fourier series unwrapping on the stratum settlement and pipeline deformation functions using a two-stage analysis method, deduced their mechanical relationship, defined the displacement delivery matrix, and proposed a simplified formula for pipeline displacement prediction based on Peck stratum settlement. Based on the Winkler elastic foundation beam model and the landslide thrust landscape distribution model, Wang Rongyou et al. [5] considered the equivalent axial force, established a simplified mechanical model of pipeline stress and deformation under the action of landscape landslides, and obtained calculated expressions such as deflection and rotation angle by solving the bending differential equation. The results show that the pipeline stress is maximum at the landslide boundary and the deformation is maximum in the middle. Based on the measured data of the shield-driven twin tunnels of Beijing Subway, Wu Fengbo et al. [6] adopted the numerical Simulation method to conduct analysis on the influence of tunnel center burial depth, horizontal pitch and other factors on surface landscape deformation. Based on the Peck Equation, they first defined the relative pitch coefficient and established the surface deformation forecasting Equation. They considered that under the condition of vertical tunnel undercrossing, the vertical deformation of pipelines can be described using a normal distribution curve, and proposed the value range of the width argument for the pipeline settlement slot. Wang Limin [7] conducted improvements on the basis of the Peck Equation. When ignoring pipeline stiffness, the deformation curve of upper pipelines caused by twin tunnel excavation was obtained. He considered that stratum settlement needs to be forecasted through superposition and fitting of two Peck curves, and the weakest location of the pipeline is where the first derivative of its deformation curvature reaches the maximum value.

The stochastic medium theory has high applicability under ultra-shallow buried conditions. Since it was proposed by Litwiniszyn J [8] and a complete system was established by Liu Baochen [9], it has been widely applied to surface settlement computation. Subsequent improvements such as the non-uniform convergence model [10] and elliptical deformation mechanism [11] have enhanced its adaptability to complex construction conditions. Based on the stochastic medium theory, Shao Zhushan [12] deduced a non-uniform convergence prediction model for different tunnel sections, optimized the model parameters using the Islands annealing genetic algorithm, and verified through the cases of the horseshoe-shaped shallow tunnel of Xi’an Metro Line 8 and the circular shallow tunnel of Mulingguan. The model prediction results are consistent with the measured data and numerical computation laws. Liu Bo et al. [13] assumed uniform convergence of the tunnel excavation section based on the Classic stochastic medium theory, proposed a parameterized non-uniform convergence pattern with superposition of 3 deformation patterns, established a surface settlement prediction model and deduced relevant prediction formulas. Back analysis combined with a shallow tunnel project in a certain section of Beijing Metro shows that the settlement slot considering non-uniform convergence is narrower and deeper, and this pattern has better prediction effect and provides empirical values of parameters. Zhang et al. [14] studied the mechanism of surface heave in ultra-shallow pipe-roof dehydrate tunneling using stochastic medium theory, proposing that when the development depth of the frozen wall exceeds the thickness of the overburden layer, the ground surface will transform into heave. Shang et al. [15], based on traditional stochastic medium theory, equated the settlement of the existing structure’s backplane to that of the overlying soil, divided the horseshoe-shaped tunnel cross-section into 8 arcing segments, considered the influence of variable cross-section of the tunnel and existing structures, treated the stratum loss at the tunnel connection as a linear transition, and introduced the concept of variable cross-section linear transition segment, thereby improving the applicability of stochastic medium theory in the excavation of variable cross-section tunnels. Wang et al. [16] established an analytical solution for forecasting surface settlement caused by shield tunnel construction in sandy cobble stratum based on stochastic medium theory. They studied the influence of tunnel geometric parameters, stratum influence angle, and volume loss on the characteristics of ground settlement through back-analysis method, and proposed an improved forecasting equation by analyzing the statistical characteristics of maximum settlement, settlement slot width, and stratum loss. In the forecasting of surface settlement induced by ultra-shallow pilot tunnel excavation using traditional stochastic medium theory, it is necessary to resolve the discontinuous settlement curve through numerical integral, which makes it difficult to directly embed into the Winkler foundation beam model to conduct pipeline deformation calculation. The settlement curve cannot be resolved analytically, and the General settlement analytical solution cannot describe the surface heave deformation caused by ultra-shallow pilot tunnels, resulting in limited accuracy of pipeline deformation forecasting. To address these critical limitations, this study proposes a novel coupled approach: integrating the polynomial-fitted continuous settlement solution (derived from stochastic medium theory) with the Winkler elastic foundation beam model. This innovation breaks the technical bottleneck of discontinuous solution embedding and establishes a dedicated calculation system for ultra-shallow burial scenarios, which has not been systematically realized in previous studies. Therefore, an efficient and accurate calculation method is required.

2. Elastic Foundation Beam Solution for the Impact of Super Shallow Buried Hidden Excavation Pilot Tunnel on Pipelines

The excavation of the super-shallow-buried concealed tunnel will cause ground surface deformation, with subsidence directly above the tunnel and significant uplift near the center. In this case, the Peck settlement trough theory is less applicable, while the random medium theory has a strong ability to describe the ground surface deformation with uplift features. A numerical solution for ground surface deformation based on the random medium theory will be established, and an approximate analytical solution will be obtained through polynomial fitting. Subsequently, it will be substituted into the elastic foundation beam model to derive the pipeline response solution.

The underground pipeline is regarded as a Winkler elastic foundation beam. The calculation assumptions are as follows: ① The settlement of the soil layer near the pipeline is obtained by the random medium theory. ② The pipeline is rigid, and the material is linear elastic, uniform, and isotropic. ③ The soil around the foundation beam is a linear elastic, uniform medium. ④ The soil around the foundation beam deforms in coordination with it and remains in contact at all times.

Notably, to overcome the discontinuity of the stochastic medium settlement solution (a key limitation of traditional methods), this study introduces a critical innovative step: 17 equidistant monitoring points are sampled along the pipeline length, and the discontinuous settlement values are converted into a continuous analytical solution via 8th-degree polynomial fitting (R² = 0.998). This continuous solution enables direct embedding into the Winkler model, realizing the effective coupling of the two theories.

The elastic foundation beam is perpendicular and orthogonal to the underpass tunnel. A coordinate system as shown in Figure 1 is established, with the point mapped from the tunnel axis to the foundation beam in the positive upward direction as the origin, the vertical downward direction as the positive z-axis, the excavation direction as the positive x-axis, and the horizontal right direction as the positive y-axis.

According to Winkler’s theory of elastic foundation beams, the settlement of foundation soil caused by tunnel excavation can be regarded as a distributed load applied to the foundation beam. The magnitude of the distributed load on the foundation beam is q(y) = Kf(y), where f(y) is the settlement of foundation soil beneath the foundation beam caused by tunnel excavation. Finally, the numerical solution of the random medium fitted by polynomial is introduced. K is the reaction modulus of the foundation bed (unit: N/m²), and the calculation formula is:

$$K={\displaystyle \frac{0.65E_{\mathrm{g}}}{1-{\mu }_g^2}}\sqrt[12]{{\displaystyle \frac{E_g{d}^4}{E_P{I}_P}}}$$

(1)

Among them, E_g is the elastic modulus of the foundation soil; μ_g is the Poisson’s ratio of the foundation soil; E_P is the elastic modulus of the foundation beam. I_P is the moment of inertia of the cross-section of the foundation beam. d is the cross-sectional thickness of the foundation beam. When deriving, take the unit length.

The differential equation of the deflection line of the pipeline micro-section is obtained as:

$$E_{\mathrm{P}}{I}_{\mathrm{P}}{\displaystyle \frac{{\mathrm{d}}^4{W}_{\mathrm{P}}}{\mathrm{d}{y}^4}}+K{W}_{\mathrm{P}}=Kf\left(y\right)$$

(2)

In the formula, W_P represents the deflection of the pipeline at y, and the meanings of the other symbols are the same as above. To simplify the derivation process, the characteristic coefficient β is introduced, which is defined as:

$$\beta =\sqrt[4]{{\displaystyle \frac{K}{4E_P{I}_P}}}$$

(3)

The general solution W_P of Equation (2) consists of the general solution W_P1 and the particular solution W_P2. The direct solution process for it is cumbersome, so the initial parameter method is introduced to assist in the solution: taking the right half of the foundation beam as the research object, and using the conditions at the left boundary of the right half beam (i.e., the origin), the relationship of the initial parameters can be derived. From mathematical knowledge, the general solution W_P1 can be obtained. By using the transformation of Euler’s formula and the hyperbolic function relationship to simplify the general solution W_P1, substituting the initial parameter expression, and then rewriting the form of the general solution W_P1 with the Krylov function, the following is obtained:

$$W_{\mathrm{P}1}=W_0{\Phi }_1\left(\beta y\right)+{\displaystyle \frac{{\theta }_0}{\beta }}{\Phi }_2\left(\beta y\right)-{\displaystyle \frac{M_0}{E_{\mathrm{P}}{I}_{\mathrm{P}}{\beta }^2}}{\Phi }_3\left(\beta y\right)-{\displaystyle \frac{Q_0}{E_{\mathrm{P}}{I}_{\mathrm{P}}{\beta }^3}}{\Phi }_4\left(\beta y\right)$$

(4)

Among them, W₀ is the initial deflection of the foundation beam at the origin, θ₀ is the initial rotation angle of the foundation beam at the origin, M₀ is the initial bending moment of the foundation beam at the origin, and Q₀ is the initial shear force of the foundation beam at the origin. The specific form of the Krylov function in Equation (4) is as follows:

$$\left\{\begin{array}{l}{\Phi }_1\left(\beta y\right)=\mathrm{ch}\left(\beta y\right)\cos \left(\beta y\right)\hfill \\ {\Phi }_2\left(\beta y\right)={\displaystyle \frac{1}{2}}\left[\mathrm{ch}\left(\beta y\right)\sin \left(\beta y\right)+\mathrm{sh}\left(\beta y\right)\cos \left(\beta y\right)\right]\hfill \\ {\Phi }_3\left(\beta y\right)={\displaystyle \frac{1}{2}}\mathrm{sh}\left(\beta y\right)\sin \left(\beta y\right)\hfill \\ {\Phi }_4\left(\beta y\right)={\displaystyle \frac{1}{4}}\left[\mathrm{ch}\left(\beta y\right)\sin \left(\beta y\right)-\mathrm{sh}\left(\beta y\right)\cos \left(\beta y\right)\right]\hfill \end{array}\right.$$

(5)

After the special solution in the form of W_P2 is obtained, the initial parameters are solved for. Distributed loads can be regarded as multiple concentrated force loads. According to mathematical knowledge, the specific solution term of each concentrated force load is known. Suppose the range of distributed loads is from the origin to section y. A micro-segment du is taken at a distance of u from the origin on the left side of Section y, and the magnitude of the load on the micro-segment is qdu. The specific solution term W_P2 caused by all distributed loads on the left side of section y is:

$$W_{\mathrm{P}2}=4\beta {\displaystyle {\int }_0^{y}f\left(u\right){\Phi }_4\left[\beta \left(y-u\right)\right]\mathrm{d}u}$$

(6)

By superimposing Equations (4) and (6), the full solution of Equation (2) can be obtained:

$$W_{\mathrm{P}}=W_{\mathrm{P}1}+W_{\mathrm{P}2}$$

(7)

Four initial parameters are calculated based on the boundary conditions as follows. Considering the left boundary condition, that is, at the origin y = 0, according to the basic principles of structural mechanics, θ₀ = 0 and Q₀ = 0 can be obtained. Regarding the right boundary condition, from a mathematical perspective, the rotation angle and deflection of the beam should be zero only when y approaches positive infinity. However, in actual engineering, at a certain distance from the tunnel axis, the rotation angle and deflection values are so small that they can be approximately regarded as zero. Assuming the effective foundation beam length within the influence range of the tunnel settlement trough is 2L, we have:

$$\left\{\begin{array}{l}{\left.0=W_{\mathrm{P}}\right|}_{y=L}\\ {\left.0={\displaystyle \frac{\mathrm{d}{W}_{\mathrm{P}}}{\mathrm{d}y}}\right|}_{y=L}\end{array}\right.$$

(8)

Solving the system of linear equations in two variables (8) yields the expressions of the remaining two initial parameters:

$$W_0={\displaystyle \frac{4\beta \left[{\Phi }_3(\beta L){\displaystyle {\int }_0^{L}f}(u){\Phi }_3\left[\beta \left(L-u\right)\right]\mathrm{d}u-{\Phi }_2(\beta L){\displaystyle {\int }_0^{L}f}(u){\Phi }_4\left[\beta \left(L-u\right)\right]\mathrm{d}u\right]}{{\Phi }_1(\beta L){\Phi }_2(\beta L)+4{\Phi }_3(\beta L){\Phi }_4(\beta L)}}$$

(9)

$$M_0={\displaystyle \frac{4{\beta }^3{E}_{\mathrm{P}}{I}_{\mathrm{P}}\left[{\Phi }_1(\beta L){\displaystyle {\int }_0^{L}f}(u){\Phi }_3\left[\beta \left(L-u\right)\right]\mathrm{d}u+4{\Phi }_4(\beta L){\displaystyle {\int }_0^{L}f}(u){\Phi }_4\left[\beta \left(L-u\right)\right]\mathrm{d}u\right]}{{\Phi }_1(\beta L){\Phi }_2(\beta L)+4{\Phi }_3(\beta L){\Phi }_4(\beta L)}}$$

(10)

During the solution process, Equations (9) and (10) are simply substituted back into the full solution expression. Among the equations, L should be selected based on the actual situation. For the settlement formula of discontinuous stochastic media, polynomial fitting is performed, and an approximate continuous settlement formula f(u) is obtained.

3. Engineering Case Analysis

3.1. Project Overview

The Xueyuan South Road Station of Beijing Metro Line 13 is planned to be constructed using the 6-way tunnel column method. The small way tunnels will be excavated under ultra-shallow burial conditions, with a soil cover thickness of 4.5 m on the arch top. The soil layers that the way tunnels pass through from top to bottom are sandy silt, clayey silt, fine silt, and clayey clay in sequence. The cross-section of the pilot tunnel is horseshoe-shaped, with a width of 4 m. The rectangular part is 3.4 m high, the circular part is 1.7 m high, and the central burial depth is 5.5 m. There is a vertical water supply pipeline buried 1 m deep spanning the pilot tunnel. The material of the pipeline is cast iron pipe, with a cut line modulus of 140 GPa. The diameter of the pipeline is 100 mm, the thickness is 6 mm, and the effective pipe length is set at 16 m. The elastic modulus of the soil layer near the pipeline is 9 MPa, and the Poisson’s ratio of the soil layer near the pipeline is taken as 0.3. To monitor the impact of construction on the pipeline, four settlement measurement points were arranged at the top of the pipeline, with an interval of 5 m between adjacent points. The spatial positions of the pilot tunnel and the pipeline, as well as the measurement points, are shown in Figure 2 step by step. The three settlement measurement points on the surface and the pipeline settlement measurement points are separated by 15 m along the direction of the pilot tunnel excavation.

3.2. Surface Settlement Analysis Based on Stochastic Media

A prediction model for ground settlement caused by ultra-shallow-buried concealed tunnel construction based on random medium [17] was established. The calculation formula for the surface deformation value W(x) is as follows:

$$W(x)={\displaystyle \underset{a}{\overset{b}{\int }}{\displaystyle \underset{c}{\overset{d}{\int }}w(x,\xi ,\eta )\mathrm{d}\xi \mathrm{d}\eta }}-{\displaystyle \underset{e^{\prime }}{\overset{f^{\prime }}{\int }}{\displaystyle \underset{g^{\prime }}{\overset{h^{\prime }}{\int }}w(x,\xi ,\eta )\mathrm{d}\xi \mathrm{d}\eta }}$$

(11)

$$w(x,\xi ,\eta )={\displaystyle \frac{\tan \beta }{\eta }}\exp \left[-{\displaystyle \frac{\pi {\tan }^2\beta }{{\eta }^2}}{(x-\xi )}^2\right]$$

(12)

a-h ‘is the integral boundary for uniform contraction and the second contraction of the horseshoe-shaped cross-section. η represents the depth coordinates of the excavated micro-soil element, and ξ represents the horizontal axis coordinates of the excavated micro-soil element. The values of the surface settlement calculation parameters are as follows: H = 5.5 m, A = 4 m, B = 3.4 m, C = 1.7 m, ΔR = 11.85 mm, tanβ = 0.45, δ = 57.65 mm. The measured values of surface settlement at a horizontal distance of 0 m along the central axis of the pilot tunnel are compared with the numerical solutions of stochastic media as shown in Figure 3. Looking from left to right, the theoretical calculation curve first drops and then rises, while the measured data shows the same trend. At a horizontal distance of −5.2 m, the measured settlement is −8.0 mm, while the theoretical calculated value is −8.2 mm. At a horizontal distance of 0 m, the measured settlement is −9.8 mm, while the theoretical calculated value is −10.0 mm. At a horizontal distance of 10.3 m, the measured settlement is −4.0 mm, while the theoretical calculated value is −4.4 mm. Overall, the theoretical calculated settlement is 0.2 to 0.4 mm greater than the measured settlement, and the calculation results are slightly conservative. Through comparison with the measured settlement data, it is found that the theoretical calculation results have relatively high accuracy and can reflect the overall variation law of settlement.

The calculation parameters for the settlement curve of the soil layer near the pipeline are as follows: H = 4.5 m, A = 4 m, B = 3.4 m, C = 1.7 m, ΔR = 11.85 mm, tanβ = 0.45, δ = 57.65 mm. After obtaining the numerical solution of the settlement curve at the pipeline height, Seventeen points were sampled at equal intervals along the horizontal distance and substituted into the numerical solution formula, and the settlement values at the corresponding positions were obtained, as shown by the red circle in Figure 4. Then, polynomial fitting was used, and the fitting curve is shown on the solid line in Figure 4. The specific form of the numerical solution transformed into the analytical solution [18,19] through polynomial fitting is as follows:

$$f\left(y\right)=3.9\times {10}^{-9}{y}^8-5.7\times {10}^{-21}{y}^7-2.1\times {10}^{-6}{y}^6+3.1\times {10}^{-17}{y}^5+1.7\times {10}^{-4}{y}^4-5.2\times {10}^{-16}{y}^3+6.8\times {10}^{-2}{y}^2+2.6\times {10}^{-14}y-1.1\times {10}^1$$

(13)

Equation (13) is a continuous analytical solution obtained by converting the numerical solution of the settlement curve at the pipeline height through polynomial fitting. The fitting data is derived from the settlement calculation values of 17 equidistant monitoring points on the distance from the central line. After multiple trials, the 8th-degree polynomial is determined by comprehensively considering the fitting accuracy and model simplicity, with a goodness of fit R² = 0.998, which can highly accurately reproduce the variation law of the original numerical solution. At the same time, the value range of the variable y (distance from the central line) is clearly defined as [−16.5 m, 16.5 m], which is slightly larger than the effective influence range of the pipeline in the engineering case [−16 m, 16 m].

3.3. Pipeline Deformation Analysis Based on Elastic Foundation Beams

Based on the stratum parameters, pipeline parameters, and the analytical expression of stratum settlement near the pipeline, a Winkler elastic foundation beam model was established. The numerical solution of settlement at a specific point on the pipeline was derived through the model. After multiple calculations, the theoretical settlement curve of the pipeline was obtained. The comparison between the theoretical value and the measured settlement of the pipeline is shown in Figure 5. The central axis of the pilot tunnel is at a horizontal distance of 0 m.

By analyzing Figure 5, it can be seen that the pipeline directly above the pilot tunnel has the greatest settlement. As the horizontal distance increases, the settlement value of the pipeline decreases. The measured maximum value of settlement is −8.6 mm, and the theoretical calculated maximum value is −10.9 mm. At a horizontal distance of 5 m, the measured value is −4.1 mm, and the theoretical calculated value is −9.2 mm. Here, the error between the measured value and the theory is the largest, with the theoretical value being 5.2 mm larger. Subsequently, as the horizontal distance increased, the error gradually decreased. At a horizontal distance of 10 m, the measured value was −2.3 mm, the theoretical calculated value was −4.2 mm, and the minimum error value was 2.0 mm. Subsequently, the theoretical settlement continued to decrease and tended to be 0 mm at a horizontal distance of 15 m, while the measured uplift here was 2.7 mm. The error between the theory and the measured value increased again. The causes of errors, especially those at the edges, are as follows: The fundamental reason is the inaccuracy of the numerical solution of stratum settlement near the pipeline derived from the stochastic medium theory. The first reason is that the parameters used in the random medium theory were not taken from the mileage where the pipeline was located. The parameters used in the random medium theory were obtained through reverse analysis of three settlement measurement points. In fact, the three settlement measurement points are 20 m apart from the pipeline measurement points along the direction of the pilot tunnel excavation. The slight difference in the characteristics of the two strata is magnified under the action of ultra-shallow burial excavation. Another reason is the lack of key data on surface settlement. Due to actual reasons, settlement measurement points were not set up near a horizontal distance of 15 m. The nearest measurement point is 5 m away. As can be seen from Figure 4, based on the three known points, the settlement at a horizontal distance of 15 m is −0.7 mm. It can be reasonably inferred that the surface deformation at this location is at least a heave of 2.7 mm or more instead of settlement. It is speculated that a bulge of 2.7 mm is actually measured for the pipeline at this location. The measured maximum settlement of −8.6 mm and the theoretical calculated maximum settlement of −10.9 mm are both less than the safety limit of 30 mm for the water supply pipeline [20]. After calculation, the MAE (Mean Absolute Error) is 3.0 mm, the RMSE (Root Mean Square Error) is 3.24 mm, indicating good overall stability of the model. The Bias is 3.0 mm, meaning the theoretical values of the model are generally 3.0 mm larger than the measured values, indicating that the settlement predicted by the theoretical model is more significant and the model is conservative. The 95% confidence interval is [1.2 mm, 4.8 mm], indicating that the error range is controllable.

As can be seen from Figure 6, starting from the center of the pipeline, the absolute value of the rotation angle increases at a constant rate with the increase in the horizontal distance. At a horizontal distance of 9.3 m, the pipeline rotation angle has a maximum value of −0.066°, and the maximum rotation angle is less than the local rotation angle limit of the water supply pipe at the welded joint [21] 0.340°. Within the horizontal distance range of 9.3 m to 14 m, the absolute value of the rotation angle decreases as the horizontal distance increases. As can be seen from Figure 7, starting from the center of the pipeline, the bending moment of the pipeline increases slowly with the increase in the horizontal distance. There is an extreme value of 40 N·m at a horizontal distance of 5 m. As the horizontal distance continues to increase, the positive bending moment value gradually decreases until it drops to zero at a horizontal distance of 9 m. When the horizontal distance continues to increase from 9 m, the negative bending moment keeps increasing. At a horizontal distance of 15.5 m, the negative bending moment of the pipeline reaches the maximum value of −137.4 N·m.

3.4. Surface Settlement Analysis Based on Finite Element Software GTS NX

The finite element software GTS NX(2022 R1) was used to simulate the pilot tunnel excavation process of Beijing Rail Transit Line 13. According to Saint-Venant’s principle, tunnel excavation affects the stratum within 3~5 times the tunnel diameter. Considering comprehensively, the model was set to 56 m in width, 24 m in height, and 30 m in length. The soil layer was modeled with solid elements and adopted the modified Mohr-Coulomb constitutive model, with different physical parameters assigned according to the geological survey report.

Before tunnel excavation, advanced small pipe grouting was used to reinforce the surrounding soil. During modeling, the effect of grouting reinforcement was converted into an enhancement of the physical and mechanical parameters of the soil in the reinforced area through equivalent calculation. According to previous research results, the elastic modulus of the undisturbed soil was doubled, and both cohesion and internal friction angle were increased by 30% on the original basis to reflect the effect of grouting reinforcement. The effective reinforcement range is considered to form a reinforcement ring of approximately 0.6–0.8 m around the excavation face.

When considering the effect of initial support, the elastic modulus of the steel mesh can be neglected compared with the grid steel arch frame, serving as a minor strength reserve. According to the principle of equivalent compressive stiffness, the grid steel frame was equivalent to concrete and superimposed with the original shotcrete strength, calculated as follows:

$$E=E_0+{\displaystyle \frac{S_{\mathrm{g}}\times E_{\mathrm{g}}}{S_0}}$$

(14)

where E is the equivalent elastic modulus of the converted initial support; E₀ is the elastic modulus of the original C25 shotcrete; S_g is the cross-sectional area of the grid steel frame; E_g is the elastic modulus of the grid steel frame; S₀ is the cross-sectional area of the original shotcrete.

The equivalent initial shotcrete and advanced small pipes are considered as linear elastic materials. The equivalent initial shotcrete adopts plate elements, the advanced small pipes use embedded trusses, and the equivalent reinforcement layer uses three-dimensional solid elements. To ensure the accuracy of the calculation results, the mesh size of the pilot tunnel and initial support is 1 m, and the stratum is divided into 2 m. The three-dimensional elements adopt mixed meshes, and the two-dimensional elements use cyclic meshes. The four sides of the model are constrained for displacement perpendicular to the side direction, the bottom surface is constrained for vertical displacement, and the upper surface is a free boundary. The model has a total of 4 elements and 6 nodes, as shown in Figure 8. When simulating construction excavation, the advanced small pipes are first installed, then the equivalent reinforcement layer is activated by simulating grouting, followed by the excavation of the outer ring soil, core soil, and lower bench, and finally the equivalent initial shotcrete is activated, with a total of six steps in one cycle, as shown in Figure 9.

The measured, theoretical model-calculated, and numerical calculated values of surface settlement are shown in Figure 10. The measured maximum surface settlement is −9.8 mm, the numerical calculated maximum is −10.2 mm, with the smallest error (the numerical calculated value is 0.4 mm larger). At a distance from the central line of −5.2 m, the measured value is −8.0 mm, and the numerical calculated value is −5.3 mm, where the error between the measured and numerical values is the largest (the numerical calculated value is 2.7 mm smaller). Subsequently, the error gradually increases with the increase of distance from the central line. At a distance from the central line of 10.3 m, the measured value is −4.0 mm, the theoretical calculated value is −0.4 mm, and the numerical calculated value is −0.4 mm, with the largest error here (the numerical calculated value is 3.6 mm smaller). The settlement results calculated by the stochastic medium theory have been consistently small compared with the measured values. The settlement results obtained by the stochastic medium theory at distances from the central line of 0 m, −5.2 m, and 10.3 m are −10.0 mm, −8.2 mm, and −4.4 mm respectively, with the largest error of 0.4 mm (the stochastic medium result is 0.4 mm larger) at a distance from the central line of 10.3 m.

The measured, theoretical model-calculated, and numerical calculated values of pipeline settlement are shown in Figure 11. The measured maximum pipeline settlement is −8.6 mm, the numerical calculated maximum is −10.9 mm, with the numerical calculated result 2.3 mm larger. At a distance from the central line of 5 m, the measured value is −4.1 mm, and the numerical calculated value is −5.3 mm, where the error between the measured and numerical values is the smallest (the numerical calculated value is 1.2 mm larger). Subsequently, the error gradually increases with the increase of distance from the central line. At a distance from the central line of 10 m, the measured value is −2.3 mm, and the numerical calculated value is −0.2 mm, with the numerical calculated result 2.1 mm smaller. Afterwards, the numerically calculated settlement decreases slowly and tends to 0 mm at a distance from the central line of 15 m, while the measured heave here is 2.7 mm, and the numerical calculation error reaches the maximum (the result is 2.7 mm smaller than the measured value).

To verify the reliability of the theoretical method, a benchmark comparison with a numerical model (GTS NX) was introduced. The quantitative statistical results of all pipeline monitoring points are shown in the

Table 1. Analysis Table of Pipeline Settlement Error.

	MAE (mm)	RMSE (mm)	R2	Bias (mm)
Theoretical Model vs. Monitored	3.0	3.24	0.89	+3.0
Numerical Model vs. Monitored	2.1	2.45	0.93	−0.6
Theoretical Model vs. Numerical Model	2.6	2.82	0.91	+3.6

:

It can be seen from the analysis that the MAE between the theoretical method and the measured values is 3.0 mm, the RMSE is 3.24 mm, and the R² reaches 0.89, indicating that the model can reproduce the pipeline settlement law with high accuracy. The Bias is +3.0 mm, meaning the theoretical values are generally larger than the measured values, reflecting a conservative characteristic that is suitable for engineering safety assessment needs. The benchmark comparison with the mature numerical model shows that the numerical model has slightly better accuracy (RMSE = 2.45 mm), but both have R² ≥ 0.89, indicating good consistency. The RMSE between the theoretical method and the numerical model is only 2.82 mm, with controllable errors, indicating that the simplified assumptions of the theoretical method (polynomial fitting of stochastic medium, Winkler elastic foundation) do not deviate from reality and do not introduce significant additional errors. In summary, the prediction accuracy of the theoretical method meets engineering requirements and has the advantage of conservatism, making it a reliable prediction tool for pipeline settlement induced by ultra-shallow pilot tunnel excavation.

4. Analysis of Pipeline Influencing Factors

4.1. Types of Strata

In practical engineering, the stratum types where pipelines are buried vary. Underground pipelines in municipal roads are often laid in miscellaneous fill layers (elastic modulus set to 2 MPa), clayey silt layers (elastic modulus set to 9 MPa), and sandy gravel layers (elastic modulus set to 50 MPa). The horizontal axis in the figure represents the horizontal distance from the tunnel axis (Figure 12). A positive bending moment indicates that the lower side of the pipeline is under tension, while a negative moment indicates that the lower side of the pipeline is under compression. The characteristics of central tension and local compression are exhibited by the bending moments of the three soil layers. Within the horizontal distance range of 0 to 6 m, the lower side of the pipeline is subjected to tension, and the bending moment values are all stable at 38 N·m. When the horizontal distance increases, the bending moment values decrease, and at a horizontal distance of 9.2 m, the bending moment drops to 0. After a horizontal distance of 10 m, the variation trends of the clayey silt layer and the sandy pebble layer are similar. Within the horizontal distance range of 9.2 to 13.7 m, the lower side of the pipeline is subjected to tension in both strata, and a maximum negative bending moment of −67.6 N·m is achieved at a horizontal distance of 12 m.

The settlement curves of the three soil layers all show the attenuation law of “large near the axis and small far the axis” (Figure 13). The pipelines at the tunnel axis are most severely disturbed and the settlement is the most significant. The settlement of the three strata is all −10.9 mm. With the increase in the horizontal distance, the settlement values gradually decrease. The settlement curves of the three strata within the horizontal distance range of 0 to 14 m almost coincide, indicating that the change in the elastic modulus of the strata has almost no impact on the settlement of the pipeline.

4.2. Pilot Tunnel Dimensions

The cross-sections of the circular arch straight wall type approach tunnels frequently used in the construction of PBA underground stations include types A, C and F (Figure 14). The cross-sectional dimensions of type A pilot tunnels are 4.1 m × 5.1 m, and they are often used for upper pilot tunnels and lower side pilot tunnels. The cross-sectional dimensions of the C-type pilot tunnel are 4.6 m × 6.5 m and it is often used for lower-level pilot tunnels. The cross-sectional dimensions of the F-type pilot tunnel are 3.0 m × 2.7 m, and it is often used for the lower pilot tunnel of the transverse passage.

As can be seen from the Figure 15, the influence trends of the A-type and F-type cross-sections on pipeline settlement are similar. The settlement is the greatest above the axis of the pilot tunnel, and gradually decreases as the horizontal distance increases. Under the influence of the A and F sections directly above the tunnel axis, the settlement of the pipeline was −8.8 mm and −5.4 mm respectively. At a horizontal distance of 15 m, the settlement of the two sections decreased to −1.6 mm and −0.8 mm respectively. The C-type cross-section has a significant impact on the pipeline settlement variation. The settlement is the greatest directly above the tunnel axis, at −18.6 mm. The settlement curve drops to the minimum at a horizontal distance of 8.3 m, with a minimum value of −2.6 mm. Then, it slightly increases again with the increase in the horizontal distance, increasing to −4.1 mm at a horizontal distance of 12.4 m, and then decreases to zero with the increase in the horizontal distance. The most significant impact on pipeline settlement and bending moment is exerted by the C-type cross-section, owing to its relatively large excavation height and wider influence range on the stratum.

Within A horizontal distance of 10 m, the influence of the A and F section guide tunnel on the pipeline bending moment is relatively small, and the bending moment values are both less than 25 N·m (Figure 16). The lower side of the pipeline is under tension, and the negative bending moment reaches the maximum value of −35 N·m at a horizontal distance of 12 m. After that, the bending moment turns positive with the increase in the horizontal distance and continues to increase. At a horizontal distance of 15 m, the bending moment value under the influence of the F-shaped section is 224 N·m, and under the influence of the A-shaped section, it is 422 N·m. The excavation height and cross-sectional dimensions of the C-shaped section are relatively large, which has the most significant impact on the bending moment of the pipeline. Within a horizontal distance of 15 m, the tensile and compressive states of the pipeline alternate. The lower side of the pipeline directly above the axis is under tension, with a bending moment value of 309 N·m. As the horizontal distance increases, the compressive state of the lower side of the pipeline gradually changes to tension. The compressive bending moment value is −250 N·m at a horizontal distance of 7.6 m. Then, the force state on the lower side of the pipeline gradually changes to tension. The tensile bending moment value is 271 N·m at a horizontal distance of 12.9 m, and the compressive bending moment value is −179 N·m at a horizontal distance of 15 m. The results are affected by both the height and width changes of the cross-sectional dimensions, but the impact of height changes is far greater than that of width changes.

4.3. Burial Depth of the Pilot Tunnel

During the construction of PBA underground stations, one or two levels of approach tunnels are often set up. The calculation conditions take three common burial depths: the center burial depth of the first level approach tunnel under ultra-shallow burial conditions is 5.5 m; under shallow burial conditions, it is 10 m; and under shallow burial conditions, it is 15 m. As can be seen from the Figure 17, under the three burial depth conditions, the settlement values of the pipeline are larger in the middle and smaller on both sides. When the burial depth of the pilot tunnel is 5.5 m, the distance between the top of the pilot tunnel and the bottom of the pipeline is relatively small, and a more significant disturbance effect is produced. The pipeline settlement is more sensitive to the change of the horizontal distance. The pipeline settlement directly above the pilot tunnel axis is −10.9 mm, and at a horizontal distance of 15 m, the pipeline settlement is −0.6 mm. Under the conditions of a pilot tunnel burial depth of 10 m and 15 m, the pipeline settlement curves change relatively gently. The pipeline settlements directly above the pilot tunnel axis are −7.1 mm and −5.1 mm respectively, and at a horizontal distance of 15 m, they are −2.4 mm and −2.7 mm respectively.

Under the three burial depth conditions, the bending moment values of the pipeline directly above the pilot tunnel axis are all positive, and the lower side of the pipeline is under tension (Figure 18). Observe and compare the bending moment values within a horizontal distance of 6.3 m. When the pilot tunnel is buried at a depth of 5.5 m, the bending moment value fluctuates slightly at 35 N·m. When the pilot tunnel is buried at depths of 10 m and 15 m, the bending moment value fluctuates slightly at 4 N·m. At a horizontal distance of 10 m, the bending moment value at the 5.5 m pilot tunnel burial depth turns negative, while the bending moment values at the 10 m and 15 m burial depths representing the shallow burial condition slightly increase towards positive values. At a horizontal distance of 11.7 m, the negative bending moments of the two reach their maximum values. The maximum negative bending moment corresponding to the ultra-shallow burial (burial depth 5.5 m) condition occurs at a relatively distant 12.6 m.

With the decrease of the pilot tunnel’s burial depth, the influence on pipeline settlement and bending moment is non-linear. Under shallow-buried conditions, the variation trends of settlement and bending moment curves with respect to the horizontal distance are similar.

4.4. Pipeline Parameters

To analyze the influence of key pipeline parameters on pipeline performance, common specifications for pipeline diameter, wall thickness, and material were selected for calculation respectively. Regarding the influence of pipeline settlement, first, for the pipeline diameter, three commonly used diameters of underground cast iron pipes were selected to illustrate the impact of their changes on the pipeline effect. The results indicate that a 50 mm change in pipeline diameter has a negligible impact on settlement. The settlement values of the three settlement curves directly above the tunnel axis are all −10.9 mm. Among them, the settlement value of the 150 mm diameter pipeline at a horizontal distance of 15 m is −0.5 mm, which only differs by 0.1 mm compared with the settlement values of the 100 mm and 150 mm diameter pipelines. The relevant changes can be referred to in Figure 19, the influence curve of pipeline diameter on settlement. Secondly, for the wall thickness of the pipeline, three commonly used thicknesses of cast iron pipes were selected for analysis. As shown in the curve of the influence of pipeline wall thickness on settlement in Figure 20, when only 1 mm of the pipeline wall thickness is changed, its influence on pipeline settlement can be ignored. Finally, for the pipeline materials, three commonly used metal materials for underground gas pipes in municipal roads were selected for calculation and analysis. Since the elastic modulus orders of magnitude of these three metal pipe materials are similar, it can be seen from the curve of the influence of pipeline materials on settlement in Figure 21 that the influence of changes in different metal pipe materials on pipeline settlement can be ignored.

In terms of the influence of pipeline bending moments, for pipeline diameters, after calculating using three commonly used diameters of underground cast iron pipes. It was found that positive extreme values are exhibited by the three pipelines with diameters of 100 mm, 150 mm, and 200 mm at a horizontal distance of 5 m, with corresponding bending moments of 41 N·m, 149 N·m, and 365 N·m respectively. There are negative extreme values at a horizontal distance of 12.3 m, with the bending moments being −70 N·m, −232 N·m, and −459 N·m respectively. Moreover, with the increase in pipeline diameter, a nonlinear increasing trend is exhibited by the pipeline’s bending moment. For specific details, please refer to the influence curve of pipeline diameter on bending moment in Figure 22. For the wall thickness of the pipeline, the three commonly used thicknesses of cast iron pipes are selected to calculate the corresponding pipeline bending moment. At the axis of the tunnel, the bending moment values corresponding to the pipeline wall thicknesses of 5 mm, 6 mm, and 7 mm are 32 N·m, 37 N·m, and 42 N·m respectively. It can be seen that the change in pipeline wall thickness exerts a slight impact on the bending moment. With the increase in wall thickness, the bending moment value increases uniformly. The relevant curves are shown in Figure 23 as the influence curve of pipeline wall thickness on bending moment. For the pipeline material, the three commonly used metal materials for underground gas pipes in municipal roads are selected to calculate the corresponding pipeline bending moment. At the axis of the pilot tunnel, the bending moment values for copper pipes, cast iron pipes, and steel pipes are 26 N·m, 37 N·m, and 56 N·m respectively. A slight impact on the bending moment is exerted by the change in pipeline material. With the increase in the pipeline’s elastic modulus, the bending moment value increases uniformly. For specific changes, please refer to the curve of the influence of pipeline material on bending moment in Figure 24.

Based on the above engineering case verification and multi-factor influence analysis, the core data and key findings of this study are summarized in

Table 2. Summary Table of Core Data.

Research Content	Core Data	Corresponding Figures
Theoretical Method Validation	Maximum theoretical settlement: −10.9 mm; measured settlement: −8.6 mm; maximum rotation angle: −0.066° (both meet safety limits)	Figure 5 and Figure 6
Error Characteristics	Maximum error: 5.1 mm (concentrated at distances from the central line of 5 m and 15 m); theoretical values are generally conservative	Figure 5
Influence of Pilot Tunnel Size	Maximum settlement of Type C pilot tunnel: −18.6 mm (with alternating bending moment); Type A: −8.8 mm; Type F: −5.4 mm	Figure 14 (cross-sections), Figure 15 and Figure 16
Influence of Pilot Tunnel Burial Depth	Axis settlement under ultra-shallow burial (5.5 m): −10.9 mm, which is approximately 56.9% higher than that under burial depth of 15 m (−5.1 mm)	Figure 17 and Figure 18
Influence of Pipeline Parameters	Extreme bending moment of pipeline with diameter 200 mm: 365 N·m (84% higher than that of 100 mm diameter); bending moment increases uniformly with the increase in wall thickness and elastic modulus	Figure 22, Figure 23 and Figure 24

, providing intuitive data support for the subsequent refinement of conclusions and engineering applications.

5. Conclusions

As shown in Table 2, the reliability of the theoretical method proposed in this study, the influence laws of various key factors, and the error characteristics are all supported by clear data. Based on these core results, the main conclusions and engineering value of the study are summarized below.

• Novelty and technical contribution: The core innovation of this study is proposing a coupled method of “stochastic medium theory discontinuous settlement solution—polynomial fitting continuous analytical solution—Winkler elastic foundation beam model” for ultra-shallow pilot tunnel projects. This method effectively resolves the long-standing technical bottleneck that traditional stochastic medium theory’s discontinuous settlement curve cannot be directly embedded into the foundation beam model, providing a dedicated and efficient analytical tool for pipeline settlement prediction under ultra-shallow burial conditions (including heave deformation, which is difficult to predict with traditional methods).
• The rationality and engineering applicability of the theoretical method have been fully verified. The coupled method of “stochastic medium theory discontinuous settlement solution—polynomial fitting continuous analytical solution—Winkler elastic foundation beam model” proposed in this paper effectively breaks through the technical bottleneck that the settlement curve of traditional stochastic medium theory is difficult to directly embed into the foundation beam model, providing an efficient analytical tool for pipeline settlement prediction in ultra-shallow pilot tunnel projects. Through the establishment of the deflection curve differential equation, the introduction of the initial parameter method, and the solution of boundary conditions, the accurate theoretical prediction of pipeline settlement is realized. Taking the ultra-shallow pilot tunnel project of Xueyuannanlu Station of Beijing Rail Transit Line 13 as an example (see Figure 5 for the comparison between measured and theoretical pipeline settlement), the results show that the variation trends of the theoretical calculated values and measured values of pipeline settlement are highly consistent. Both the maximum theoretical settlement value (−10.9 mm) and the measured value (−8.6 mm) are much smaller than the safety limit of 30 mm, and the maximum rotation angle (−0.066°) is lower than the limit of 0.340° for welded joints, fully proving the reliability of this method in actual engineering.
• The error characteristics, causes, and applicability of the engineering example are clear. The errors between the theoretical calculations and the measured values are mainly concentrated at a distance from the central line of 5 m and the pipeline edge area (distance from the central line of 25 m), with a maximum error of 5.1 mm. The causes of the errors are directly related to the range of parameter values of the stochastic medium theory and the layout of monitoring points in the edge area. The parameters are taken from the surface settlement data 20 m away from the pipeline monitoring points, and the differences in stratum characteristics are amplified under ultra-shallow burial conditions. In addition, the lack of key settlement monitoring points in the edge area leads to fitting deviations. The theoretical calculated values are generally larger than the measured values, for example, the maximum theoretical settlement value (−10.9 mm) is greater than the measured value (−8.6 mm). This conservative characteristic is consistent with the research conclusion of Shao Zhushan et al. [12] on tunnel settlement prediction based on stochastic medium theory, and the core control indicators (maximum settlement, rotation angle) all meet the safety limits specified in related specifications [20,21], which can provide reliable safety redundancy for pipeline safety assessment in ultra-shallow pilot tunnel projects.
• The laws of influencing factors of pipeline deformation are verified by experiments and supported by literature. Multi-factor analysis shows that the pilot tunnel size is the core influencing factor (see Figure 14). Due to its large excavation height and cross-sectional size, the Type C pilot tunnel has the most significant influence on pipeline settlement and bending moment, with a maximum settlement of −18.6 mm and alternating tension and compression of bending moment (see Figure 15 and Figure 16 for relevant influence curves). This law is consistent with the core conclusion of “excavation scale dominates stratum disturbance degree” in the study of surface deformation of ultra-shallow tunnels by Zhang et al. [14]. The smaller the burial depth of the pilot tunnel, the more obvious the disturbance. Under the ultra-shallow burial (5.5 m) condition, the pipeline settlement is more sensitive to changes in distance from the central line. This characteristic is directly verified by the measured data of Xueyuannanlu Station of Beijing Rail Transit Line 13 (the settlement directly above the axis is increased by about 56.9% compared with the condition of burial depth of 15 m). The influence laws of pipeline parameters and stratum elastic modulus are consistent with the pipeline deformation research results based on the Winkler model by Wang Haitao et al. [4], further proving the universality of the laws.
• Research Limitations: Limited coverage of spatial working conditions: This study only focuses on the typical working condition of “tunnel vertically undercrossing the pipeline,” and does not cover complex spatial layout forms such as parallel or oblique crossing between the tunnel and the pipeline, resulting in certain limitations in the applicable scenarios of the model. Insufficient validation sample size: Due to the high construction difficulty and monitoring cost of ultra-shallow pilot tunnel projects, publicly available measured data and engineering cases are relatively scarce. This study only relies on one engineering example of Beijing Rail Transit Line 13 for validation, and the limited sample size may affect the universality of the conclusions. Room for improvement in prediction accuracy: The errors between the theoretical calculations and measured values in the pipeline edge areas (e.g., distances from the central line of 5 m and 15 m) are relatively large (maximum error of 5.1 mm), mainly affected by the indirectness of parameter inversion of the stochastic medium theory, the lack of edge monitoring points, and stratum heterogeneity. The accuracy loss caused by these factors has not been completely eliminated.
• Subsequent Research Plan: Expand the scope of applicable working conditions: For complex crossing forms such as parallel and oblique crossing between the tunnel and the pipeline, supplement and establish corresponding mechanical models, improve the pipeline deformation prediction method under different spatial working conditions, and expand the engineering application scenarios of the research. Expand the support of validation data: Increase the sample size of model validation by collecting more measured data of ultra-shallow pilot tunnel projects and conducting supplementary numerical simulation calculations, further improving the universality and reliability of the conclusions. Optimize prediction accuracy: Address the error problem in edge areas, improve the parameter inversion method of the stochastic medium theory, introduce stratum heterogeneity correction coefficients, and improve the layout scheme of monitoring points to reduce fitting deviations caused by data lack, thereby improving the overall prediction accuracy.