Analysis of the Buried Pipeline Response Induced by Twin Tunneling Using the Generalized Hermite Spectral Method

Abstract

For the sustainability of economic, ecological and social development, the safety of infrastructure, including buried pipelines, is extremely important. Undercrossing tunneling can compromise the safety of buried pipelines because of deformations, cracks and dislocations, which can result in wasted resources, environmental pollution and economic losses. Therefore, it is important to assess the pipeline response accurately during tunnel excavation. This paper proposes a generalized Hermite spectral solution to estimate the pipeline response induced by twin tunneling. The proposed solution is formulated by a truncated series of Hermite functions and it is available in an unbounded domain. On the basis of the two-stage analysis method, a general formula for calculating the soil greenfield displacement induced by twin tunneling is first derived using the superposition principle. To obtain the final solution, the soil greenfield displacement and pipeline displacement are expanded using two truncated series of Hermite functions, and the governing differential equation of pipeline displacement is subsequently simplified into a linear algebraic system. After solving this system, a general solution for calculating pipeline displacement is formulated. Then, the convergence of the developed solution is proven, and its validity is verified against existing theoretical solutions and centrifuge test results. The effects of the truncated series number and its scaling factor are investigated. Finally, parametric studies are conducted to discuss pipeline responses induced by twin tunneling.

1. Introduction

In urban areas, buried pipelines are commonly utilized to transport water, sewage, oil, natural gas and other materials, and they play extremely important roles in the sustainability of economic, ecological and social development. Because of space limitations, most buried pipelines are often constructed in a complex environment, and they are vulnerable to surface loading, foundation-pit excavation and tunnel excavations [1,2,3,4,5]. With the rapid growth of underground transportation systems in big cities, more and more tunnels have been excavated beneath buried pipelines in the past few decades. Tunnel excavations inevitably disturb the surrounding soils and bring about additional stress on buried pipelines and further significantly affect their deformation and mechanical behavior, which may cause damage or even accidents (i.e., as breakage or bursting [6,7,8,9]). Therefore, for the sustainability and safety of buried pipelines, it is of critical importance to accurately estimate the buried pipeline response during tunnel excavations.

In the community of geotechnical engineering, it is a key subject of great interest to estimate the buried pipeline response induced by undercrossing tunneling, and on this topic, a great deal of research has been conducted over the last few decades. Attewell et al. [10] first proposed an analytical solution to evaluate the buried pipeline response induced by single tunneling. They introduced the Winkler foundation model to describe the pipeline–soil interaction and the greenfield displacement caused by a single tunneling was assumed as a Gaussian curve. Afterwards, Klar et al. [3,11,12] and Vorster et al. [13] developed a series of elastic continuum solutions for calculating the buried pipeline response induced by undercrossing tunneling. Wang et al. [14] derived a novel analytical solution to predict the buried pipeline response during undercrossing tunneling, and they divided the buried pipeline into uplift and downward segments with different subgrade modulus. By introducing four virtual nodes to describe the boundary conditions at two truncated ends, Zhang et al. [15,16] developed a finite difference solution to calculate the deformation of a buried pipeline. These solutions used the Winkler foundation model to describe the pipeline–soil interaction, and they ignored the continuum of the soil stratum. To address this issue, Lin and Huang [17,18] introduced the Pasternak model to study the buried pipeline response induced by undercrossing tunneling. Similarly, Lin et al. [19] put forward a finite difference solution to compute the deformation of two overlying pipelines caused by undercrossing tunneling. These aforementioned studies generated valuable knowledge about the tunneling effects of undercrossing tunnels on buried pipelines, but they mainly focused on the scenario of single tunneling. In practical situations, most urban tunnels are generally constructed in pairs because of space limitations. For this problem, many scholars have conducted comprehensive studies, including physical model experiments, numerical simulations and theoretical analyses, in recent years. Ma et al. [20,21] developed a series of centrifuge tests to study the influences of tunnel depth, soil loss ratio, and excavation sequence on the buried pipeline response. Yu et al. [22] performed a comprehensive numerical simulation encompassing 8640 conditions to investigate the mechanical and deformation behavior of buried pipeline that was adjacent to undercrossing twin tunnels. Compared with physical model experiments and numerical simulations, theoretical solutions are commonly more convenient and can provide guidance more efficiently for the primary designs of actual projects [3,19,23]. Based on the two-stage analysis method, Liu et al. [24] used the superposition principle to evaluate the soil greenfield displacement induced by twin tunneling and further derived an analytical solution to compute the deformation of buried pipeline. Recently, Gan et al. [25] used the Timoshenko–Winkler model to develop an analytical solution to assess the buried pipeline response caused by undercrossing twin tunnels.

In general, the aforementioned theoretical works can be classified into two different kinds: analytical method and finite difference method. The analytical method can only account for the case with a simple foundation model and simple external action simultaneously, while there is usually no analytical solution for other cases. The finite difference method can be available for general cases with a complex foundation model and/or complex external action, but it needs to divide the pipeline into a series of very small segments, which generally results in a tremendous amount of requirement on computation resources and time. Moreover, these two kinds of methods are only workable for a finite long pipeline that is artificially truncated with fictitious boundary conditions. In fact, the buried pipeline is commonly much longer than the acting region of additional stress arising from undercrossing tunneling; thus, it seems to be more reasonable to take the pipeline as infinite long for developing a solution to assess its response [1]. Different from the analytical method and finite difference method, the generalized Hermite spectral method formulates a solution of the differential equation using a truncated series of Hermite functions, which are mutually orthogonal and workable in an unbounded domain; thus, it can be available for estimating the buried pipeline response in the whole region. In particular, compared with the finite difference method, there is no need to divide the buried pipeline into a series of small segments, which brings a significant advantage in saving computation resources and time, i.e., much more efficiently [26,27]. Using this method, many scholars have developed a series of solutions in an unbounded domain for some classical differential equations stemming from various fields of mathematical sciences, physical sciences, chemical sciences and engineering sciences, such as the Fokker–Planck equation [28,29] and reaction–diffusion equation [30]. Its efficiency and availability in unbounded domains have been widely tested and powerfully validated in these applications. To date, however, no effort has been devoted to introducing it to formulate a general solution for assessing the buried pipeline response induced by tunneling.

Through introducing the generalized Hermite spectral method, this paper attempts to provide a novel general solution to assess the response of buried pipelines caused by twin tunneling to ensure its sustainability and safety. On the basis of the two-stage analysis method, the tunneling effect on the buried pipeline can be simplified into an external load that depends on the soil greenfield displacement. Accordingly, this paper first formulates the basic expression for calculating the soil greenfield displacement induced by twin tunneling using the superposition principle and the Gaussian curve. Subsequently, the governing differential equation of buried pipeline displacement is presented using Euler–Bernoulli beam theory and the Pasternak foundation model. Then, using the generalized Hermite spectral method, the presented governing differential equation is changed into a linear algebraic system with respect to the displacement vector of the buried pipeline at collocation points, and a general solution is formulated to estimate the buried pipeline response in the whole domain. The convergence of this solution is further proven, and its validity is verified against existing theoretical solutions and centrifuge test results. Finally, to illustrate the buried pipeline response induced by twin tunneling, parametric studies are conducted for deep investigation.

2. Mathematical Model

2.1. Problem Definition

A buried pipeline with twin undercrossing tunnels is schematically shown in Figure 1. The pipeline is buried at a depth of $z_{\mathrm{p}}$, and its diameter is $D_{\mathrm{p}}$. Twin tunnels are simply considered to be undercrossing the pipeline perpendicularly. Their radii are $R_1$ and $R_2$, and their depth and spacing are $H_{\mathrm{t}}$ and $L$, respectively. In addition, the distance between the twin tunnels and the buried pipeline is $d_{\mathrm{n}}$. The excavations of these tunnels inevitably disturb the surrounding soil and further lead to soil displacement, which causes additional stress and deformation of the buried pipeline. To assess the pipeline response effectively, a two-stage analysis method is adopted. This method divides the study into two steps: the soil greenfield displacement induced by tunnel excavations is first calculated; and second, the pipeline response is computed based on a reasonable beam model and pipeline-soil interaction model (i.e., foundation model). In this paper, the buried pipeline is represented by a continuous Euler–Bernoulli beam, and the soil-pipeline interaction is modeled by the Pasternak model. For ease of mathematical derivation, all the assumptions applied to investigate the buried pipeline response caused by tunneling are retained, and the prominent ones are listed as follows [9,13,16,19,31]:

(1) The tunnels and pipeline are all horizontally buried in a homogeneous isotropic elastic soil stratum.

(2) The pipeline is elastic, and its presence does not affect tunneling or soil greenfield displacement.

(3) The pipeline and surrounding soil are always tightly in contact, and their interaction behavior is linear elastic.

(4) Only the vertical deformation of the buried pipeline is considered.

The aforementioned assumptions have been widely adopted to develop theoretical solutions for assessing buried pipeline responses induced by tunneling.

2.2. Greenfield Displacement Induced by Twin Tunneling

Peck [32] investigated a large number of actual tunneling projects in San Francisco, Oakland and Berkeley, and he pointed out that the profile of soil greenfield surface settlement induced by single tunneling can be simplified as the following Gaussian curve:

$$S\left(x\right)=S_{\max }\exp \left(-\frac{x^2}{2i^2}\right),$$

(1)

where $S\left(x\right)$ is the settlement at the position of $x$ ($x$ is the lateral distance from tunnel axis to any point on its settlement trough), $S_{\max }$ is the maximum settlement at the tunnel axis, and $i$ is the lateral distance from tunnel axis to the inflection point of its settlement curve. Numerous field observations have proven that the subsurface settlement due to single tunneling can also be described using the above Gaussian curve after modifying $i$ as [33,34]:

$$i=K_{\mathrm{s}}\left(H_{\mathrm{t}}-\lambda z_{\mathrm{p}}\right),$$

(2)

where $H_{\mathrm{t}}$ is the tunnel depth, $z_{\mathrm{p}}$ is the pipeline depth, $K_{\mathrm{s}}$ is the trough width parameter at the ground surface, and $\lambda$ is an empirical parameter that reflects the changing rate of $K_{\mathrm{s}}$ with depth.

With the rapid growth of traffic systems in big cities, more and more twin tunnels are excavated beneath buried pipelines. Therefore, many studies have been conducted to predict the soil greenfield settlement caused by twin tunneling based on the superposition principle and Gaussian curve [32,35,36,37]. Due to the sequential excavations of twin tunnels, the first tunneling inevitably disturbs the surrounding soil. As a result, the settlement caused by second tunneling is usually different from that induced by first tunneling. To address this issue, Liu et al. [24] proposed a modified expression for evaluating the subsurface settlement induced by twin tunneling:

$$\begin{array}{ll}S\left(x\right)& =S_1\left(x\right)+S_2\left(x\right)\\ & =-\frac{A_1{V}_l}{i\sqrt{2\pi }}{\mathrm{e}}^{\left[-\frac{1}{2i^2}{\left(x-\frac{L}{2}\right)}^2\right]}-\left(1+\eta -\frac{\eta }{ni}\left|x-\frac{L}{2}\right|\right)\frac{A_2{V}_l}{i\sqrt{2\pi }}{\mathrm{e}}^{\left[-\frac{1}{2i^2}{\left(x+\frac{L}{2}\right)}^2\right]},\end{array}$$

(3)

where $S_1\left(x\right)$ and $S_2\left(x\right)$ are the subsurface settlements induced by the first and second tunneling, respectively; $A_1$ and $A_2$ are the cross-section areas of the first and second tunnels, respectively; $V_l$ is the volume loss ratio; $\eta$ is a modification factor of subsurface settlement induced by the second tunneling for considering the soil disturbing degree from the first tunneling, and $n$ is a multiplier to modify $i$. Take the value of $\frac{1}{ni}\left|x-\frac{1}{2}L\right|$ as one if it is greater than one.

2.3. Governing Differential Equation

Using the Pasternak model to characterize the pipeline–soil interaction behavior, the differential equation governing the deformation of a buried pipeline can be expressed as [15,24,38]:

$$E_{\mathrm{p}}{I}_{\mathrm{p}}\frac{d^{4}w(x)}{d{x}^4}+D_{\mathrm{p}}kw(x)-D_{\mathrm{p}}{G}_{\mathrm{p}}\frac{d^{2}w(x)}{d{x}^2}=D_{\mathrm{p}}q(x),$$

(4)

where $w\left(x\right)$ is the vertical displacement of the buried pipeline, $q\left(x\right)$ is the additional load acting on the buried pipeline, $E_{\mathrm{p}}{I}_{\mathrm{p}}$ is the equivalent bending stiffness of the buried pipeline, $k$ is the reaction coefficient of the subgrade, and $G_{\mathrm{p}}$ is the shear stiffness of the subgrade. For a buried pipeline, the reaction coefficient of subgrade $k$ can be given by [11]:

$$k=\frac{6E_{\mathrm{s}}}{i},$$

(5)

and the shear stiffness of subgrade $G_{\mathrm{p}}$ can be evaluated from [39]:

$$G_{\mathrm{p}}=\frac{E_{\mathrm{s}}h}{6\left(1+v\right)},$$

(6)

where $E_{\mathrm{s}}$ is the elastic modulus of soil, $v$ is Poisson’s ratio of soil, and h is the thickness of the shear layer, which can be simply taken as $6D_{\mathrm{p}}$ [15,40].

The additional load $q\left(x\right)$ acting on the buried pipeline that would be induced by the soil greenfield displacement $S\left(x\right)$ can be obtained from [24,41]:

$$q\left(x\right)=kS\left(x\right)-G_{\mathrm{p}}\frac{d^{2}S\left(x\right)}{d{x}^2}.$$

(7)

Substituting Equation (7) into Equation (4) results in:

$$\frac{d^{4}w\left(x\right)}{d{x}^4}+\frac{D_{\mathrm{p}}k}{E_{\mathrm{p}}{I}_{\mathrm{p}}}w\left(x\right)-\frac{D_{\mathrm{p}}{G}_{\mathrm{p}}}{E_{\mathrm{p}}{I}_{\mathrm{p}}}\frac{d^{2}w\left(x\right)}{d{x}^2}=\frac{D_{\mathrm{p}}k}{E_{\mathrm{p}}{I}_{\mathrm{p}}}S\left(x\right)-\frac{D_{\mathrm{p}}{G}_{\mathrm{p}}}{E_{\mathrm{p}}{I}_{\mathrm{p}}}\frac{d^{2}S\left(x\right)}{d{x}^2}.$$

(8)

In general, the buried pipeline is long enough, and it is free at both ends that approach infinity; thus, its boundary conditions are [16]:

$${\left.w\left(x\right)\right|}_{x=\pm \infty }=0, {\left.\frac{dw\left(x\right)}{dx}\right|}_{x=\pm \infty }=0.$$

(9)

3. Solution Formulation

3.1. Generalized Hermite Spectral Method

The Hermite functions are defined by [42]:

$${\hat{H}}_n\left(\xi \right)=\frac{1}{{\pi }^{1/4}\sqrt{2^{n}n!}}{e}^{-{\xi }^2/2}{H}_n\left(\xi \right), n\ge 0,$$

(10)

where $H_n(\xi )$ is Hermite polynomial of degree $n$ with respect to univariate $\xi$, and it satisfies the following recurrence relations [43]:

$$\begin{array}{l}\begin{array}{cc}{H}_0\left(\xi \right)=1,& H_1\left(\xi \right)=2\xi \end{array},\\ H_{n+1}\left(\xi \right)=2\xi H_n\left(\xi \right)-2n{H}_{n-1}\left(\xi \right),\hspace{1em}n\ge 1.\end{array}$$

(11)

Substituting Equation (11) into Equation (10) produces

$$\begin{array}{c}{\hat{H}}_{n+1}\left(\xi \right)=\xi \sqrt{\frac{2}{n+1}}{\hat{H}}_n\left(\xi \right)-\sqrt{\frac{n}{n+1}}{\hat{H}}_{n-1}\left(\xi \right),\hspace{1em}n\ge 1,\\ {\hat{H}}_0\left(\xi \right)={\pi }^{-1/4}{e}^{-{\xi }^2/2},\hspace{1em}{\hat{H}}_1\left(\xi \right)=\sqrt{2}{\pi }^{-1/4}\xi e^{-{\xi }^2/2}.\end{array}$$

(12)

An arbitrary function $u\left(\xi \right)$ can be expanded as the following truncated series of Hermite functions [42]:

$$u\left(\xi \right)={\displaystyle \sum _{n=0}^N{\tilde{u}}_n}{\hat{H}}_n\left(\xi \right),$$

(13)

where ${\left\{{\tilde{u}}_n\right\}}_{n=0}^N$ are expansion coefficients of function $u\left(\xi \right)$, and they can be computed by:

$${\tilde{u}}_n={\displaystyle \sum _{j=0}^{N}u\left({\xi }_j\right)}{\hat{H}}_n\left({\xi }_j\right){\hat{\omega }}_j, 0\le n\le N,$$

(14)

where ${\left\{{\xi }_j\right\}}_{j=0}^N$ are the zeros of ${\hat{H}}_{N+1}\left(\xi \right)$, which are called collocation points of the generalized Hermite spectral method. ${\left\{{\hat{\omega }}_j\right\}}_{j=0}^N$ are the quadrature weights. Herein, the zeros ${\left\{{\xi }_j\right\}}_{j=0}^N$ are equal to the eigenvalues of the below symmetric tridiagonal matrix:

$$A_{N+1}=\left[\begin{array}{ccccc}{a}_0& \sqrt{b_1}& & & \\ \sqrt{b_1}& a_1& \sqrt{b_2}& & \\ & \ddots & \ddots & \ddots & \\ & & \sqrt{b_{N-1}}& a_{N-1}& \sqrt{b_N}\\ & & & \sqrt{b_N}& a_N\end{array}\right],$$

(15)

where $a_j=0,\hspace{1em}0\le j\le N;\hspace{1em}{b}_j=j/2,\hspace{1em}1\le j\le N.$

The quadrature weights ${\left\{{\hat{\omega }}_j\right\}}_{j=0}^N$ are given by:

$${\hat{\omega }}_j=\frac{1}{\left(N+1\right){\hat{H}}_N^2\left({\xi }_j\right)}, 0\le j\le N.$$

(16)

The $m$ times differentiation of Equation (13) with respect to $\xi$ is:

$$u^{\left(m\right)}\left(\xi \right)={\displaystyle \sum _{n=0}^N{\tilde{u}}_n}{\hat{H}}_n^{\left(m\right)}\left(\xi \right).$$

(17)

Let us denote $u={\left\{u\left({\xi }_0\right),u\left({\xi }_1\right),\dots ,u\left({\xi }_N\right)\right\}}^{\mathrm{T}}$ and $u^{\left(m\right)}={\left\{u^{\left(m\right)}\left({\xi }_0\right),u^{\left(m\right)}\left({\xi }_1\right),\dots ,u^{\left(m\right)}\left({\xi }_N\right)\right\}}^{\mathrm{T}}$, then Equation (17) can be rewritten as

$$u^{\left(m\right)}=\stackrel{m}{\overbrace{DD\dots D}}u=D^{m}u,\hspace{1em}m\ge 1,$$

(18)

where $D$ is the first-order differentiation matrix of the generalized Hermite spectral method, and its elements are computed by [42]:

$${\hat{d}}_{kj}=\left\{\begin{array}{ll}\frac{{\hat{H}}_N\left({\xi }_k\right)}{{\hat{H}}_N\left({\xi }_j\right)}\frac{1}{{\xi }_k-{\xi }_j},& \mathrm{if} k\ne j\\ 0,& \mathrm{if} k=j\end{array}\right..$$

(19)

3.2. Solution for Buried Pipeline Response

For a problem with solution decaying at infinity, there is an effective interval outside of which the solution is negligible, and the collocation points that fall outside this interval are essentially wasted. Thus, in order to enhance the approximating performance of the generalized Hermite spectral method, it usually needs to determine a proper scaling factor to make the smallest and largest collocation points at or close to the endpoints of this interval. Accordingly, by defining a scaling factor $\beta$ and letting $x=\beta \xi$, $\psi \left(\xi \right)=w\left(x\right)$, $\mathsf{\Gamma }\left(\xi \right)=S\left(x\right)$, then Equation (8) can be rewritten as

$${\beta }^{-4}\frac{d^4\psi \left(\xi \right)}{d{\xi }^4}+\frac{D_{\mathrm{p}}k}{E_{\mathrm{p}}{I}_{\mathrm{p}}}\psi \left(\xi \right)-{\beta }^{-2}\frac{D_{\mathrm{p}}{G}_{\mathrm{p}}}{E_{\mathrm{p}}{I}_{\mathrm{p}}}\frac{d^2\psi \left(\xi \right)}{d{\xi }^2}=\frac{D_{\mathrm{p}}k}{E_{\mathrm{p}}{I}_{\mathrm{p}}}\mathsf{\Gamma }\left(\xi \right)-{\beta }^{-2}\frac{D_{\mathrm{p}}{G}_{\mathrm{p}}}{E_{\mathrm{p}}{I}_{\mathrm{p}}}\frac{d^2\mathsf{\Gamma }\left(\xi \right)}{d{\xi }^2}.$$

(20)

Similarly, functions $\psi \left(\xi \right)$ and $\mathsf{\Gamma }\left(\xi \right)$ can be expanded as the following truncated series of Hermite functions:

$$\psi \left(\xi \right)={\displaystyle \sum _{n=0}^N{\tilde{\psi }}_n}{\hat{H}}_n\left(\xi \right),$$

(21)

$$\mathsf{\Gamma }\left(\xi \right)={\displaystyle \sum _{n=0}^N{\tilde{\mathsf{\Gamma }}}_n}{\hat{H}}_n\left(\xi \right),$$

(22)

where ${\left\{{\tilde{\psi }}_n\right\}}_{n=0}^N$ and ${\left\{{\tilde{\mathsf{\Gamma }}}_n\right\}}_{n=0}^N$ are the expansion coefficients of functions $\psi \left(\xi \right)$ and $\mathsf{\Gamma }\left(\xi \right)$, and they can be expressed as:

$${\tilde{\psi }}_n={\displaystyle \sum _{j=0}^N\psi \left({\xi }_j\right)}{\hat{H}}_n\left({\xi }_j\right){\hat{\omega }}_j, 0\le n\le N,$$

(23)

$${\tilde{\mathsf{\Gamma }}}_n={\displaystyle \sum _{j=0}^N\mathsf{\Gamma }\left({\xi }_j\right)}{\hat{H}}_n\left({\xi }_j\right){\hat{\omega }}_j, 0\le n\le N,$$

(24)

It should be noted that the expansion coefficients ${\left\{{\tilde{\mathsf{\Gamma }}}_n\right\}}_{n=0}^N$ can be directly computed from the known function $\mathsf{\Gamma }\left(\xi \right)$ (i.e., the soil greenfield displacement $S\left(x\right)$), and the expansion coefficients ${\left\{{\tilde{\psi }}_n\right\}}_{n=0}^N$ are the those to be determined for calculating the displacement of the buried pipeline.

Let $\psi ={\left\{\psi \left({\xi }_0\right),\psi \left({\xi }_1\right),\dots ,\psi \left({\xi }_N\right)\right\}}^{\mathrm{T}}$ and $\mathsf{\Gamma }={\left\{\mathsf{\Gamma }\left({\xi }_0\right),\mathsf{\Gamma }\left({\xi }_1\right),\dots ,\mathsf{\Gamma }\left({\xi }_N\right)\right\}}^{\mathrm{T}}$, the governing differential Equation (20) can be further simplified into the linear algebraic system below:

$$\left({\beta }^{-4}{D}^4+\frac{D_{\mathrm{p}}k}{E_{\mathrm{p}}{I}_{\mathrm{p}}}I-\frac{D_{\mathrm{p}}{G}_{\mathrm{p}}}{E_{\mathrm{p}}{I}_{\mathrm{p}}}{\beta }^{-2}{D}^2\right)\psi =\left(\frac{D_{\mathrm{p}}k}{E_{\mathrm{p}}{I}_{\mathrm{p}}}I-\frac{D_{\mathrm{p}}{G}_{\mathrm{p}}}{E_{\mathrm{p}}{I}_{\mathrm{p}}}{\beta }^{-2}{D}^2\right)\mathsf{\Gamma }.$$

(25)

From Equation (25), $\psi$ can be solved as:

$$\psi ={\left({\beta }^{-4}{D}^4+\frac{D_{\mathrm{p}}k}{E_{\mathrm{p}}{I}_{\mathrm{p}}}I-\frac{D_{\mathrm{p}}{G}_{\mathrm{p}}}{E_{\mathrm{p}}{I}_{\mathrm{p}}}{\beta }^{-2}{D}^2\right)}^{-1}\left(\frac{D_{\mathrm{p}}k}{E_{\mathrm{p}}{I}_{\mathrm{p}}}I-\frac{D_{\mathrm{p}}{G}_{\mathrm{p}}}{E_{\mathrm{p}}{I}_{\mathrm{p}}}{\beta }^{-2}{D}^2\right)\mathsf{\Gamma }.$$

(26)

Then, substituting solution $\psi$ into Equation (23) can result in the expansion coefficients ${\left\{{\tilde{\psi }}_n\right\}}_{n=0}^N$, and further using these expansion coefficients, the vertical displacement of the buried pipeline along longitudinal direction can be obtained from Equation (21) as

$$w\left(x\right)={\displaystyle \sum _{n=0}^N{\tilde{\psi }}_n}{\hat{H}}_n\left({\beta }^{-1}x\right).$$

(27)

4. Solution Verification

In this section, two worked examples are presented to prove the convergence of the proposed solution and also to verify its exactness against the existing theoretical solution developed by Liu et al. [24] and the centrifuge test result presented by Ma et al. [20]. The effects of the scaling factor $\beta$ and truncated series number $N$ are illustrated and discussed.

4.1. Worked Example I

The first worked example comes from a buried pipeline project in London and the Piccadilly Line tunnels were excavated through it [44]. The external diameter, wall thickness and depth of the buried pipeline are 4.1 m, 0.15 m and 13 m, respectively, and its equivalent bending stiffness is 80 $\mathrm{GPa}\cdot {\mathrm{m}}^4$. The diameters, depth and spacing of the Piccadilly Line tunnels are 9.1 m, 26 m and 23 m, respectively. The elastic modulus of London clay is estimated using the formula of $E_{\mathrm{s}}=\left(10+5.2z_{\mathrm{p}}\right) \mathrm{MPa}$, which was given by [45], and its Poisson’s ratio is assumed to be 0.3. The other input parameters for this worked example are the same as those used by Liu et al. [24], and they are listed in

Table 1. Input parameters of worked example I.

${\mathit{A}}_1$$\left({\mathbf{m}}^2\right)$	${\mathit{A}}_2$$\left({\mathbf{m}}^2\right)$	${\mathit{V}}_{\mathit{l}}$$\left(\mathbf{\%}\right)$	${\mathit{E}}_{\mathit{s}}$$\left(\mathbf{MPa}\right)$	${\mathit{K}}_{\mathit{s}}$	$\mathit{\lambda }$	$\mathit{\eta }$
65	65	1.2	77.6	0.5	0.65	1.5

.

To prove the convergence of the proposed solution, the effects of the scaling factor $\beta$ and the truncated series number $N$ are first illustrated and discussed. For twin tunnel excavations, the induced greenfield displacement can be evaluated according to Equation (3), and for determining the buried pipeline response, Equation (3) should be expanded as Equation (22) with Hermite functions. The approximation performance of Equation (22) is the fundamental premise to obtain a reliable result for the pipeline response. Without loss of generality, herein, the overall relative error at all the collocation points is adopted to assess the approximation performance of Equation (22). For five given scaling factors $\beta$ (i.e., $\beta$ = 6, 8, 10, 12, 14), Figure 2a shows the overall relative error from Equation (22) with the increasing truncated series. It can be seen that for all the scenarios with different scaling factors, the overall relative error decreases rapidly with the increase in the adopted truncated series. For the scenario of $\beta$ = 6, the overall relative error decreases to below 5% when the adopted truncated series exceeds 20, while for the other cases, just 10 truncated series are needed to reduce the overall relative error below 5%. Similarly, taking the truncated series as 10, 20, 30 and 40 in turn, Figure 2b illustrates the overall relative error from Equation (22) with the increase of the scaling factor. As observed, with the increase in the scaling factor, the overall relative error initially decreases to a minimum value and then gradually increases. That is, there is a proper scaling factor that can minimize the calculated error from Equation (22). These findings demonstrate that the developed solution can be convergent with the increase of the truncated series, and relatively few truncated series (i.e., $N$ = 20) can lead to a reliable result with high accuracy. The value of the scaling factor has a significant influence on the approximation performance of the developed solution, and a proper scaling factor (i.e., $\beta$ = 10) can produce the final result more efficiently with fewer truncated series. The influence of the scaling factor gradually decreases with the increase of the adopted truncated series, and once the adopted number of truncated series is relatively large (i.e., $N$ ≥ 30), the influence of the scaling factor is very small and can even be negligible.

Taking the scaling factor $\beta$ as 10 and truncated series $N$ as 10, 20 and 30 in turn, the soil greenfield displacement of this worked example can be calculated according to Equation (22). Figure 3a shows the soil greenfield displacements calculated by Equation (22) and Equation (3), and the calculation errors between them are depicted in Figure 3b. It can be seen that the approximation result with 10 truncated series shows a slight difference from that given by Equation (3), while the truncated series is taken as 20 and 30; the corresponding approximation results are in excellent agreement with those given by Equation (3). Figure 3b illustrates that the computed error is relatively obvious in the influencing region of twin tunnels, and it decays rapidly away from this influencing region.

Moreover, for this worked example, the vertical displacement of the buried pipeline is calculated by the developed generalized Hermite spectral solution (i.e., Equation (27)). Figure 4 shows the vertical displacement of the buried pipeline along its longitudinal direction in the whole domain. As a comparison, the results evaluated from the theoretical solution proposed by Liu et al. [24] are also plotted in this figure. As seen, only using 10 truncated series, the developed can provide a reliable result with a slight difference from that given by Liu et al. [24], while the truncated series increase to 20 or 30. The result from the proposed solution is in excellent agreement with that of Liu et al. [24] in the whole domain. Thus, the developed generalized Hermite spectral solution can effectively estimate the buried pipeline response, and it is available in the whole domain (i.e., unbounded domain).

4.2. Worked Example II

The second worked example comes from the centrifuge test that was performed by Ma et al. [20]. The buried pipeline was modeled by an aluminum tube with an equivalent bending stiffness of 10.68 $\mathrm{GPa}\cdot {\mathrm{m}}^4$. Its external diameter, wall thickness and depth are 31.75 mm, 2.08 mm and 95.875 mm, respectively. The diameters, depth and spacing of the twin-tunnel model are 100 mm, 450 mm and 200 mm, respectively. According to the scale factors summarized by Ma et al. [20], the dimensions of the modeled pipeline are equivalent to 1.905 m, 0.126 m and 5.753 m in the prototype, respectively. The dimensions of the modeled tunnels are equivalent to 6 m, 27 m and 12 m in the prototype, respectively. The Poisson’s ratio of Toyoura sand that was adopted in this centrifuge test is assumed to be 0.3, and its elastic modulus is 650 MPa, which was given by Yamashita et al. [46]. The other input parameters for this worked example are the same as those used by Ma et al. [20], and they are listed in

Table 2. Input parameters of worked example II.

${\mathit{A}}_1$$\left({\mathbf{m}}^2\right)$	${\mathit{A}}_2$$\left({\mathbf{m}}^2\right)$	${\mathit{V}}_{\mathit{l}}$$\left(\mathbf{\%}\right)$	${\mathit{K}}_{\mathit{s}}$	$\mathit{\lambda }$	$\mathit{\eta }$
28.27	28.27	2	0.45	1.124	0.0(0.3)

. To show the superiority of the modification factor $\eta$ on characterizing the tunneling effect from the first tunnel but also on the latter one, two cases of $\eta =0$ and $\eta =0.3$ are discussed. The case of $\eta =0$ indicates that the tunneling effect from the first tunnel is not considered, which is the same as the theoretical solution of Ma et al. [20]; while the case of $\eta =0.3$ is presented to consider this tunneling effect, and the value of the modification factor is evaluated from the measured pipeline response in the centrifuge test.

For this worked example, the effects of the scaling factor $\beta$ and the truncated series number $N$ are also presented to demonstrate the convergence of the proposed solution. Taking the scaling factor $\beta$ as 8, 9, 10, 11 and 12, Figure 5a shows the overall relative error from Equation (22) with the increasing truncated series for the case of $\eta =0.3$. As observed, with the increase of the adopted truncated series, the overall relative error reduces rapidly, and it generally becomes lower than 5% when the adopted truncated series exceeds five. Similarly, taking the truncated series as 4, 6, 8 and 10 in turn, Figure 5b shows the overall relative error from Equation (22) with the increase in scaling factor. As the scaling factor increases, it can be obviously seen that the overall relative error from Equation (22) initially decreases to a minimum value and then gradually increases; that is, there is an optimal scaling factor (i.e., $\beta$ = 12.5) that can produce the final result more efficiently with fewer truncated series. Once the number of adopted truncated series exceeds 10, the value of the scaling factor shows an insignificant influence on the approximation performance (i.e., the overall relative error) of Equation (22). These findings are the same as those found in the first worked example, and they demonstrate that the developed solution is convergent. Relatively few truncated series can provide a reliable result with high accuracy, and a proper scaling factor can present the final result more efficiently with much fewer truncated series (i.e., much less computation resource).

Taking scaling factor $\beta$ as 12.5, truncated series $N$ as five, and modification factor $\eta$ is taken as 0.0 and 0.3, in turn, the soil greenfield displacement can be computed according to Equation (22) for this worked example. Figure 6a shows the soil greenfield displacements calculated by Equation (22) and Equation (3), and the calculation errors between them are shown in Figure 6b. As observed, the approximation results from Equation (22) show excellent agreements with those given by Equation (3). Figure 6b indicates that the approximating error from Equation (22) is relatively large at the influencing region of twin tunnels, and it decreases rapidly away from this region. These findings are the same as those found in the first worked example, and they indicate that the proposed solution can obtain the soil greenfield displacement effectively with several truncated series (i.e., collocation points) and a proper scaling factor.

Furthermore, the vertical displacement of the buried pipeline for this worked example is calculated by the developed generalized Hermite spectral solution (i.e., Equation (27)). Figure 7 illustrates the vertical displacement distribution of the buried pipeline along its longitudinal direction in the whole domain. For the case where the modification factor $\eta$ equals zero, which does not consider the disturbance effect from the first tunnel on the second one, the calculated value from the proposed solution coincides with the theoretical result of Ma et al. [20]. However, these calculated values are smaller than the measured results; that is, the pipeline response induced by twin tunnel excavations is underestimated. For the case in which the modification factor $\eta$ equals 0.3, the developed solution predicts the pipeline response more accurately, and the difference between the calculated values from Equation (27) and the measured results given by Ma et al. [20] are very small. This finding indicates that the disturbance effect from the first tunneling on the second one is usually significant for twin tunneling, and that the modification factor $\eta$ can characterize this disturbance reliably. Thus, the developed solution, including the modification factor, seems to be more reasonable to evaluate the buried pipeline response due to twin tunneling.

5. Parametric Studies

Based on the developed generalized Hermite spectral solution, this section presents a series of parametric studies to investigate the deformation behavior of buried pipelines induced by twin tunneling. The influences of the spacing of twin tunnels, the distance between twin tunnels and buried pipelines, the equivalent bending stiffness of buried pipelines, and the modification factor of soil greenfield displacement are discussed. In this section, the adopted calculation parameters are the same as the first worked example presented in the previous section.

5.1. Influence of the Spacing of Twin Tunnels

Taking the spacing of twin tunnels is 10 m, 15 m, 20 m, 25 m, and 30 m in sequence, the vertical displacement of the buried pipeline induced by twin tunneling can be computed from the proposed solution (i.e., Equation (27)). For these cases, Figure 8a illustrates the displacement distribution of the buried pipeline in the whole domain. It can be observed that with the increase in spacing of twin tunnels, the distribution pattern of pipeline displacement changes from a “V” type to a “W” type, while the corresponding deformation region broadens. As a result, the displacement of the buried pipeline becomes much more uniform with a smaller value in the main influencing region of twin tunneling, which is very helpful in controlling the deformation of the buried pipeline and keeping it safely. This phenomenon benefits from the overlapping effect of twin tunneling and the slighter disturbance of the first tunneling in the scenario with larger spacing, which can reduce the soil greenfield displacement. In Figure 8b, the maximum displacement of the buried pipeline is depicted, which indicates that the maximum displacement of the buried pipeline gradually reduces with the increase in the spacing of twin tunnels. Once the spacing is large enough (i.e., larger than 25 m), the spacing change shows less impact on buried pipeline displacement, indicating that the overlapping effect of twin tunneling becomes very small or even negligible. In this scenario, the twin tunnels can be simply treated as two single ones.

5.2. Influence of the Distance between Tunnels and Pipeline

Figure 9a illustrates the displacement distributions of buried pipeline due to twin tunneling with five different distances between twin tunnels and buried pipeline, and the depth of buried pipeline is unchanged. As observed, when this distance is relatively small (e.g., 7 m), the distribution pattern of buried pipeline displacement appears to be “W” type because the influencing regions of twin tunneling overlap slightly in this situation, which leads to two distinct peaks on the soil greenfield displacement curve, similar to two single tunnels. As the distance between the twin tunnels and buried pipeline increases, these influencing regions overlap more and more obviously, which results in greater soil greenfield displacement, and its peak appears in the middle region of twin tunnels. As a result, the pipeline displacement becomes larger, and its distribution pattern changes into a “V” type. Figure 9b shows the corresponding variation in the maximum displacement when the distance between the twin tunnels and buried pipeline increases. It can be seen that the maximum displacement of the buried pipeline firstly decreases with this distance increase, and then it gradually increases with a decreasing rate. Once this distance exceeds a certain value (i.e., 17 m), its change shows a slight effect on pipeline displacement, indicating that the influencing regions of twin tunnels almost overlap.

5.3. Influence of the Equivalent Bending Stiffness of Buried Pipeline

Taking the equivalent bending stiffness of the buried pipeline as $20 \mathrm{GPa}\cdot {\mathrm{m}}^4$, $40 \mathrm{GPa}\cdot {\mathrm{m}}^4$, $80 \mathrm{GPa}\cdot {\mathrm{m}}^4$, $160 \mathrm{GPa}\cdot {\mathrm{m}}^4$ and $320 \mathrm{GPa}\cdot {\mathrm{m}}^4$. Figure 10a presents their displacement distributions due to twin tunneling, which shows that with the increase in the equivalent bending stiffness of the buried pipeline, its displacement distribution pattern changes from a “W” type to a “V” type, and meanwhile the maximum displacement moves toward the middle of twin tunnels. In the case of a small equivalent bending stiffness, the buried pipeline is very flexible, and it usually deforms in accordance with the pattern of external actions, i.e., the “W” pattern of soil greenfield displacement induced by twin tunneling; while in the case of a large equivalent bending stiffness, the buried pipeline is relatively rigid, thus its deformation will be different from external actions, i.e., the “V” pattern differs from soil greenfield displacement. Figure 10b demonstrates the change in the maximum displacement of the buried pipeline when its equivalent bending stiffness increases. As observed, the maximum displacement of a buried pipeline gradually decreases with the increase in its equivalent bending stiffness; that is, for a stiffer buried pipeline, the induced displacement by undercrossing tunnels is smaller.

5.4. Influence of the Modification Factor of Greenfield Displacement

The modification factor of greenfield displacement reflects the degree of soil disturbance caused by the first tunneling. Taking it as 0, 0.5, 1, 1.5 and 2, Figure 11a depicts the displacement distribution of the buried pipeline induced by twin tunneling. It can be observed that this factor has a significant impact on buried pipeline displacement. As it increases, the pipeline displacement gradually increases, and its distribution pattern changes from a “W” type to a “V” type. This phenomenon happens because a larger modification factor means greater soil disturbance from the first tunneling, which can further enlarge the soil greenfield displacement due to the second tunneling. In Figure 11b, the corresponding variation of the maximum displacement is displayed with an increase in the modification factor. As observed, for twin undercrossing tunneling, the maximum displacement of the buried pipeline enlarges gradually with the modification factor increases.

6. Conclusions

Estimating the response induced by undercrossing tunneling effectively is significant for the sustainability and safety of buried pipelines. To address this issue, this paper has developed a novel solution for assessing the displacement of buried pipelines induced by twin tunneling. The solution technique adopts a generalized Hermine spectral method, which formulates the solution in an unbounded domain using the truncated series of Hermite functions. The convergence of the developed solution has been discussed and its validity has been verified against the existing theoretical solution and centrifuge test result. Parametric studies have been conducted to deeply investigate the response behavior of buried pipelines induced by twin tunneling. The main conclusions can be drawn as follows:

(a) The developed solution shows excellent agreement with the existing theoretical solution and centrifuge test result, and it is available in the whole domain (i.e., unbounded domain). The scaling factor and truncated series number have obvious influences on the approximation performance of this solution. A greater number of truncated series produces the final result more accurately, and a proper scaling factor can minimize the calculation error. With the increase in adopted truncated series, the developed solution can converge rapidly, and once the number of adopted truncated series reaches a certain value, the scaling factor shows an insignificant influence on convergence.

(b) The calculation parameters, including the spacing of twin tunnels, the distance between twin tunnels and buried pipeline, the equivalent bending stiffness of buried pipeline and the modification factor of soil greenfield displacement, show significant influences on the response behavior of buried pipeline. With the decrease in the spacing of twin tunnels and the increases in the other three parameters, the distribution pattern of pipeline displacement usually changes from a “W” type to a “V” type, indicating that the overlapping effect of twin tunneling turns from slight to relatively obvious.

(c) As tunnel spacing enlarges, pipeline displacement gradually decreases and distributes more uniformly in the main influencing region of twin tunneling. This is beneficial for controlling pipeline deformation and keeping it safely. The increase in the distance between twin tunnels and buried pipelines may intensify the overlapping effect of twin tunneling and further lead to a larger pipeline displacement. For a flexible pipeline (i.e., small equivalent bending stiffness), it deforms following the pattern of soil greenfield displacement, while for a stiff pipeline (i.e., large equivalent bending stiffness), it shows a better capability to adjust the deformation and also reduce the overall response. As the modification factor increases, i.e., the soil disturbance from the first tunneling intensifies, the pipeline displacement gradually enlarges.