# Ground Displacements due to the Deformations of Shallow Tunnels with Arbitrary Cross Sections in Soft Ground

Authors: Qiongfang Zhang; Kang Cheng; Cungang Lin; Yadong Lou; Panpan Guo; Xiaolu Gan
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

Nowadays, a huge number of shield-driven tunnels with noncircular cross sections are constructed in urban areas all around the world. However, the ground displacements associated with tunneling still form a difficult issue, especially for noncircular tunnels. In this study, an analytical solution is derived to estimate the ground displacements induced by the deformations of shallow noncircular tunnels in soft ground. First, a solution for the stresses and displacements around a deep tunnel in a full plane is formulated by imposing a specified convergence pattern over the cavity boundary. Subsequently, this solution is validated using finite element simulations in a case study of an elliptical tunnel with four different convergence patterns. Afterward, the solution in the full plane is extended to a half plane using the virtual image technique to estimate the ground displacements around shallow tunnels. The solution is also validated using finite element simulations.


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## Body

## 1. Introduction

In the last twenty years, a variety of special-shaped shields have been developed to build tunnels with many different shapes and cross sections [1–7]. Compared with the most widely used circular tunnels, noncircular tunnels are believed to have the advantage of making full use of excavated space [8]. Thus, they have been more and more frequently employed in urban areas for transportation development.During the construction of shallow tunnels in soft ground, one critical concern is the ground movements associated with tunneling. At present, there have been several methods available for predicting tunneling-induced ground movements, including empirical methods, analytical solutions, model tests, and numerical simulations [9]. Although numerical simulations are generally powerful and elaborate, analytical solutions are still necessary, as they are less time-consuming and easier for practical applications compared with other methods [6, 10]. In particular, when abundant experiences are obtained from field observations and model tests, analytical solutions with empirical modifications can provide as many accurate results as numerical simulations.Among analytical solutions, those derived from elastic theories make up important categories. These elastic solutions can generally be formulated by a combination of complex variable methods [11] and some well-established theorems, such as the Cauchy’s integral theorem [12], Laurent’s theorem [13], and theorems on conformal mapping [12]. The elastic solutions for generated stresses and displacements around deep tunnels in rocks are mostly derived in infinite planes by imposing a specified loading distribution over boundaries at certain distances from tunnels [14–17]. However, since the tectonic stresses acting on shallow openings are not as high as those acting on deep openings, such pressure-controlled derivation is generally not applicable to shallow tunnels [15]. Instead, a displacement-controlled method, in which, for example, a specified convergence pattern is imposed on the cavity of a shallow tunnel and is employed for estimating the soil stresses and displacements around shallow tunnels [18–24]. Sagaseta [22] regarded the excavation of a tunnel as the radial convergence of a small circle to the tunnel axis. The authors of [23] developed the study of [22] by considering the effect of Poisson’s ratio of soil and an elliptical convergence pattern around a tunnel boundary. The authors of [19, 20] derived elastic solutions for soil displacements around both deep and shallow tunnels in clay by assuming four different convergence patterns. Pinto and Whittle [21] pointed out that tunneling-induced ground settlements depend on convergence patterns that are both uniformly radial and elliptical. Overall, it has been proven by previous studies that the ground displacements due to the excavations of shallow circular tunnels can be reliably calculated under appropriately assumed convergence patterns around cavities.More and more tunnels with noncircular cross sections are constructed worldwide [25–29]. However, most of the abovementioned solutions for tunneling-induced soil stresses and displacements are related to circular tunnels, giving sparse attention to noncircular tunnels. Wang et al. [30] derived an elastic solution for shallow tunnels with arbitrary cross sections in rock and stiff soil using the Schwartz alternating method [31]. It is obtained by superimposing an infinite plane solution [32] and a half-plane solution [33]. However, this solution is only applicable to deep tunnels.In this study, an elastic solution is derived to estimate the ground displacements around shallow tunnels in soft ground. First of all, an elastic solution in a full plane is formulated to calculate the ground displacements around deep tunnels with arbitrary cross sections. Afterward, it is employed in the deep tunnels of several commonly used shapes. Furthermore, the solution is extended to a half-plane for shallow tunnels using the virtual image technique [22].


## 2. Elastic Solutions in a Full Plane

### 2.1. Formulations

In this section, the issue to be solved is the soil stresses and displacements around a deep tunnel with an arbitrary cross section in an infinite plane. It is assumed that the cavity occupied by the tunnel is vacuum and that the surrounding soil is homogeneous, isotropic, and linearly elastic. Elastic solutions for soil stresses and displacements can be obtained by following the steps as follows:(1)
Conformally mapping the glyphs in the original domain onto the regular graphs in the mapped plane(2)
Translating the conditions of the displacement boundary or stress boundary into a complex variable domain using the conformal mapping technique(3)
Getting the unknown parameters of the stress function while assuming that the stress of a point at infinity is zero(4)
Using the available theorems, such as Cauchy’s integral theorem and Laurent’s theorem, to simplify the stress function


#### 2.1.1. Complex Variable Function

In the absence of body forces, the stresses around a cavity can be solved using the Airy stress functionF as(1)∂4F∂y4+2∂4F∂x2∂y2+∂4F∂x4=0,where x and y denote the horizontal and vertical axes, respectively, in a Cartesian coordinate system, as shown in Figure 1.Figure 1
Thez-plane.The equilibrium can be satisfied ifF meets the following conditions [12]:(2)σx=∂2F∂y2,σy=∂2F∂x2,τxy=−∂2F∂x∂y,where σx and σy are the normal stresses in the x and y directions, respectively, and τxy is the shear stress in the x-y plane.The displacements around the cavity can be expressed by the two harmonic functions,φz and ψz, as follows [34]:(3)2Gux+iuy=κφz−zφ′z¯−ψz¯,where ux and uy are the displacements in the x and y directions, respectively; G is the shear modulus of the soil; z is a complex variable in the z-plane, z=x+iy; the constant κ is related to Poisson’s ratio v by κ=3−4v for the plane strain problems and κ=3−v/1+v for the plane stress problems, respectively; ν is Poisson’s ratio of soil; and φz and ψz are functions with respect to the variable z, which can be determined by giving the displacement boundary conditions around the tunnel cavity. φ′z¯ and ψz¯ are the conjugates of φ′z¯ and ψz¯, respectively.Afterward, the stresses around the cavity can be expressed in the form of complex functions as follows:(4)σx+σy=4Reφ′z,σy−σx+2iτxy=2z¯φ″z+ψ′z,where z¯ is the conjugate of z and Re[.] denotes the real part of a generic complex variable [⋅].


#### 2.1.2. Conformal Mapping

A tunnel with an arbitrary cross section (ellipse, circle, oval, and rectangle) in thez-plane is transformed into a unit circular hole in the ζ-plane, as depicted in Figure 2. The employment of the mapping technique z=wζ yieldFigure 2
Conformal mapping of the exterior of an unlined tunnel of any shape onto the exterior of a unit circle.(5)φz=φwζ=φζ,ψz=ψwζ=ψζ.For a 2-dimensional (2D) elastic problem, the displacements and stresses are expressed in terms of two complex variable functions,φ(z) and ψ(z), with z = x + iy and i=−1. By introducing conformal mapping and equation (5), the generated stresses and displacements around the tunnel cavity can be expressed as follows:(6)2Gux+iuy=κφζ−wζw′ζ¯φ′ζ¯−ψζ,¯(7)σx+σy=4Reφ′ζw′ζ,σy−σx+2iτxy=2w′ζwζ¯φ′ζw′ζ′+ψ′ζ,where wζ¯ is the conjugate of the complex function wζ, ζ is a complex variable in the ζ-plane, ζ=ξ+iη=ρeiϑ, ξ and η are variables of the Cartesian coordinates system in the mapped plane, and ρ and ϑ are variables of the polar coordinates system in the mapped plane.From (7), the stress components can be obtained as follows:(8)τxy=Im1w′ζwζ¯φ′ζw′ζ′+ψ′ζ,σx=2Reφ′ζw′ζ−Re1w′ζwζ¯φ′ζw′ζ′+ψ′ζ,σx=2Reφ′ζw′ζ−Re1w′ζwζ¯φ′ζw′ζ′+ψ′ζ,where Im[⋅] denotes the imaginary component of the generic complex variable [⋅].Since (6) is analytic in the mapped ζ-plane, it can be expanded in the Laurent series of σ using ζ=ρeiϑ=ρσ as follows:(9)2Gux+iuy=∑−∞∞Akeinθ=∑−∞∞Anσn=gσ,where σ denotes a point on the unit circle in the ζ-plane. Therefore, the coefficients Ak are determined by means of the recursive relations derived from the given boundary conditions.The displacement components can be obtained as follows:(10)ux=Regσ=12GRe∑−∞∞Anσn,uy=Imgσ=12GIm∑−∞∞Anσn.As shown by Muskhelishvili, the stress boundary condition in the phase plane can be expressed as follows:(11)φζ+wζwζ¯φ′ζ¯+ψζ¯=i∫fx+fyds,where fx and fy are the surface stresses in the x and y directions, respectively, where they act on the elliptical cavity wall.It can be verified [34] that the potentials can be written in the form as follows:(12)φζ=18π1−vFx¯+iFy¯lnζ+Bwζ+φ0ζ,(13)ψζ=−3−4v8π1−vFx¯−iFy¯lnζ+B′+iC′wζ+φ0ζ,where Fx¯+iFy¯ is the resultant force applied to the boundary (which in this case is equal to zero). The variables B, B’ + iC’, and B’ − iC’ can be written as(14)B=σ1+σ24,(15)B′+iC′=−12σ1−σ2e−2iα,where σ1 and σ2 are the principal stresses at infinity and α is the angle between the maximum principal direction and x-direction.It is assumed that the stresses vanished at infinity and on the tunnel boundary. Thus,(16)Fx¯+iFy¯=0,B′+iC′=0.By substituting (16) into (12) and (13),(17)φζ=φ0ζ,ψζ=ψ0ζ,where the functions φ0ζ and ψ0ζ are single-valued and analytic in the ζ-plane, including the point at infinity.According to equations (6), (9), and (17),(18)gσ=∑−∞∞Anσn=3−4vφ0σ−wσw′σ¯φ0′σ¯−ψ0σ¯.By taking the conjugate, then(19)gσ¯=∑−∞∞Anσ−n¯=3−4vφ0σ¯−wσ¯w′σφ0′σ−ψ0σ.Equations19 and 20 are multiplied by dσ/2πiσ−ζ and integrated along the unit circle in the ζ-plane:(20)12πi3−4v∮φ0σσ−ζdσ−12πi∮wσw′σ¯φ0′σ¯σ−ζdσ−12πi∮ψ0σ¯σ−ζdσ=12πi∮gσσ−ζdσ,(21)12πi3−4v∮φ0σ¯σ−ζdσ−12πi∮wσ¯w′σφ0′σσ−ζdσ−12πi∮ψ0σσ−ζdσ=12πi∮gσ¯σ−ζdσ.According to the properties of the Cauchy integral,wσ¯,w′σ,andφ0′σ are analytic functions in the unit circle. Then, wσ¯/w′σφ0′σ is analytic in the unit circle. It can be concluded that the second and third terms on the left-hand side of (17) are zero and that the first term on the left-hand side of (18) is zero. Thus, it can be concluded that(22)φ0ζ=−13−4v12πi∮gσσ−ζdσ.By combining (9) and (23),(23)φ0ζ=13−4v∑−∞0Anζn.A similar process could be performed to findψ0ζ, where An are the only unknown constants in the two complex variable functions.


#### 2.1.3. Tunnels with Different Cross Sections

To find a solution for the present problem, we first consider the transformation of the tunnels with arbitrary cross sections, such as ellipses, circles, oval shapes, and squares, in thez-plane into a unit circular hole in the ζ-plane. The transformation function [35] is assumed as(24)z=wς=Rς+∑n=1Nanς−n,ς=ρeiθ,where R is a real number that refers to the cross-section size and an is a general complex coefficient satisfying an<1/n [36]. Inverse mapping is analytical, single-valued, and nonzero in the exterior part of the curve.In many instances, it is assumed that the physical domain possessesp symmetry axes and then yields(25)z=wς=Rς+∑n=1Na1pnς1−pn.The tunnel cavity assigns the prescribed radial displacement along its boundary, and the central position of the curve remains unchanged after deformation. It remains axisymmetric along thex-axis and y-axis. After deformation, the curve can be expressed as(26)Z=wcζ=Rcζ+∑n=1Nbnζ−n.The total displacementuk at the tunnel boundary can be expressed as(27)u=Z−z,=ux+iuy,=Rc−Rζ+∑n=1NRcbn−Ranζ−n.Compared with (10), the coefficient Ak is obtained as follows:(28)A1=2GRc−RAn=0n≥2An=2GRcbk−Rakn≤−1.For a cylindrical cavity of radiusr in isotropic soil, the boundary condition is solved by mapping onto a circle of a unit radius:(29)z=rζ.When undergoing uniform convergenceuε, the deformed tunnel boundary is expressed as follows:(30)Z=r−uεζ.Considering an infinite plane containing an elliptical tunnel cavity with a major axis 2a1 and a minor axis 2b1, the transformation function became(31)z=wζ,=R0ζ+mζ−1,where R0=a1+b1/2andm=a1−b1/a1+b1.Heller et al. [37] provide a mapping function for a rectangular opening of unit width and height, K, using the Schwarz–Christoffel integral:(32)z=Rζ+1−K241ζ−K2241ζ3−K21−K21601ζ5−K21−3K2+K48961ζ7+⋯.Manh et al. [16] propose that, to get a smaller error for the elastic fields at the corner of a rectangular opening, at least 10 terms must be used in the conformal mapping functions. Once an approximate polynomial mapping for the origin and deformed shape of a cavity boundary is known, the elastic solution can be determined using the analytical method proposed in the paper. Generally, tunnel contours comprise complex curves. To generalize these studies, it is reasonable to simplify complicated tunnels as unlined tunnels with elliptical outlines [38].


#### 2.1.4. Different Deformation Patterns

It is of practical significance to determine the coefficients of the approximate polynomial mapping function after deformation. Taking the elliptical tunnel as an example, four convergence patterns in Figure3 are assumed to describe the displacement on the boundary of the elliptical tunnel:Figure 3
Boundary conditions of the given displacement.(33)B.C.−1:uc=−u0sinθ+tcos2θ,(34)B.C.−2:uc=−u01+sinθ−1−tcos2θ,(35)B.C.−3:uc=−u01+sinθ−1−t2cos2θ,(36)B.C.−4:uc=−u045+3sinθ−5−2tcos2θ,1≤t≤a1b1.It should be noted that the given convergence patterns are reduced to the boundary conditions given by [19] when t=1 and a1=b1.On the boundary of the elliptical cavity in thez-plane,(37)zc=xc+iyc=acosθ+ibsinθz¯c=xc−iyc=acosθ−ibsinθ⇒cosθ=zc+zc¯2a,sinθ=zc−zc¯2bi.On the boundary of the unit circle in theζ-plane,(38)ζ=eiϑ=σ,ζ¯=e−iϑ=σ−1,(39)zc=Rζc+mζc=Rσ+mσ−1,zc¯=Rζc¯+mζc¯=Rσ−1+mσ.By substituting (35) into (3),(40)cosθ=R1+mσ+σ−12a,sinθ=R1−mσ−σ−12bi,(41)Fx,y=2Gux20+iuy20=2Guccosθ+icosθ.By substituting (31)–(34) and (38) into (39), the displacement can be expressed as a polynomial of σk. The constant Ak is obtained through a comparison with (9). By substituting Ak in (8), the stress components are found.


### 2.2. Application to an Elliptical Tunnel in a Full Plane

Considering that an infinite plane contains an elliptical tunnel cavity with a major axis 2a1 and a minor axis 2b1, as illustrated in Figure 1, a uniform radial displacement (B.C.1, t=1) is assumed as the boundary condition for the elliptical tunnel cavity. The initial elliptical tunnel cavity is converged without altering the ratio of the semimajor axis to the semiminor axis of the elliptical tunnel cavity.(42)a1a2=b1b2,=ρ1ρ2,=1k,where a2 and b2 correspond to the current semimajor and semiminor axes of the elliptical tunnel cavity, respectively; ρ1,ρ2 denote the angular length of the elliptical tunnel at any point before and after the deformation in the polar coordinate system, respectively; and k is the convergence ratio.At the boundary of the elliptical tunnel cavity,(43)ux0=−1−k,x=−1−kz+z¯2=−1−kRm+1σ+m+1σ−12,uy0=−1−k,y=−1−kz−z¯2=−1−kR1−mσ−1−mσ−12.By comparing (23) and (25), there are only two Fourier coefficients for the complex variable function:(44)A1=2G1−kR,A−1=2G1−kRm,Ak=0k>1,k<−1.


#### 2.2.1. Comparison of the Solutions for the Elliptical and Circular Tunnels

The ratio of the semimajor and semiminor axes of the initial elliptical cavity (a/b) is 5/4, and the soil Poisson’s ratio equals 0.5. The elastic solution of the elliptical tunnel cavity and the circumscribed circular tunnel cavity expansion shown in Figure 4 are obtained using the analytical method. The orthoradial displacement and shearing stress caused by the radial displacement of the circumscribed circular tunnel cavity and elliptical tunnel cavity along the semimajor and semiminor axis directions of the elliptical tunnel cavity are zero.Figure 4
Current elliptical tunnel cavity and circumscribed circular tunnel cavity before and after the radial displacement at the boundary of the original tunnel cavity.Figure5 shows the radial displacement variation of the soil in relation to the distance from the center of the cavities, normalized by the radius of the circumscribed circular tunnel cavity. The analytical results of the radial displacement along the major and minor axes of the elliptical tunnel cavity are compared with the results of the circumscribed circular tunnel cavity.Figure 5
Distribution of the radial displacement of the soil due to the radial displacement at the boundaries of the cavities.It can be seen that the radial displacement of the soil around the elliptical tunnel cavity along the major and minor axes and circumscribed circular tunnel cavity decreases with the increase in the distance from the cavity’s center and that it eventually tends to zero. The radial displacement of the soil along the major axis of the elliptical tunnel cavity is equal to that of the circumscribed circular tunnel cavity.Figure6 shows the radial and orthoradial stresses of the soil, which are caused by the radial displacement of the elliptical tunnel cavity and circumscribed circular tunnel cavity in relation to the distance from the center of the cavities. The magnitude of the radial stress of the soil along the semimajor axis of the elliptical tunnel cavity is larger than the radial stress of the soil along the radial direction of the circumscribed circular. The radial stresses along the semiminor axis of the elliptical tunnel cavity are minimum. The magnitude of the soil orthoradial stress induced by the radial displacement of the circumscribed circular tunnel cavity is larger than the soil orthoradial stress caused by the elliptical tunnel cavity. As the distance from the cavity center increased, the soil radial and orthoradial stresses around the elliptical tunnel cavity and circumscribed circular tunnel cavity tend to zero.Figure 6
Distribution of the stress of the soil elements due to the radial displacement at the boundaries of the cavities.Whena=4m,G=2000kPa,andk=0.92 are considered, the magnitudes of the soil stress and displacement around the cavities are very small σρ<1kPa,σθ<1kPa,τρθ<1kPa,uρ<0.010mm, at ∼50 m from the cavity center. That is, the influence radius of the convergence of the elliptical cavity is ∼20 times that of the semimajor axis.


#### 2.2.2. Elastic Solutions of the Soil around an Elliptical Tunnel Cavity Using the Analytical and Finite Element Method Methods

To validate the accuracy of the results, the solution is compared with a finite element method (FEM) calculation using the ABAQUS 2D software. The FEM model size is1000×1000m, and the boundary elements were infinite elements. Since the whole model is symmetric about the origin point and the x, y axes, one-quarter of the whole model is selected for comparison. The soil’s stress and displacement around the elliptical ellipse are compared and analyzed. Figures 7 and 8 show the contour of the normalized stresses and displacements of the soil from the analytical solution with the FEM solution. The selected parameters of the elliptical tunnel cavity and soil are as follows:(45)a1H=547,k=0.92,G=2000kPa,v=0.4.where a is the semimajor axis, k is the convergence ratio, G is the shear modulus, and v is the soil Poisson’s ratio.Figure 7
Contours of the normalized stresses and displacements in the case of (a)/(b) = 2 as derived by the theoretical method and ABAQUS 2D. (a) Shear stress of the soil. (b) Soil stress in thex-direction. (c) Soil stress in the y-direction. (d) Soil displacement in the x-direction. (e) Soil displacement in the y-direction.
(a)(b)(c)(d)(e)Figure 8
Contours of the normalized stresses and displacements in the case of the elliptical tunnel with different ellipticity values. (a) Shearing stress of soil. (b) Soil stress in thex-direction. (c) Soil displacement in the x-direction. (d) Soil displacement in the y-direction. (e) Soil stress in the y-direction.
(a)(b)(c)(d)(e)It can be seen from Figures7(a)–7(e) that the soil shearing stress in the directions of the x-axis and the y-axis is zero. There are some differences between the values of the vertical stresses and displacements because the boundary condition of the ABAQUS model is finite and fixed at the bottom. The soil stress and displacement calculated using the complex function method and FEM are approximately equal, which verifies the correctness of the theoretical method in this paper.


#### 2.2.3. Elastic Solutions of the Soil around the Elliptical Tunnel Cavity with Different Ellipticity Values

Figures8(a)–8(e) show the soil stress and displacement caused by the convergence of the elliptical cavities with different ellipticity values. From the figures, it can be seen that the ellipticity of the cavity decreased, the soil stress in the x-axis direction of the soil increased, and that the soil stress and displacement in the y-axis direction decreases. Also, the soil displacement in the x-axis direction can be seen.


## 2.1. Formulations

In this section, the issue to be solved is the soil stresses and displacements around a deep tunnel with an arbitrary cross section in an infinite plane. It is assumed that the cavity occupied by the tunnel is vacuum and that the surrounding soil is homogeneous, isotropic, and linearly elastic. Elastic solutions for soil stresses and displacements can be obtained by following the steps as follows:(1)
Conformally mapping the glyphs in the original domain onto the regular graphs in the mapped plane(2)
Translating the conditions of the displacement boundary or stress boundary into a complex variable domain using the conformal mapping technique(3)
Getting the unknown parameters of the stress function while assuming that the stress of a point at infinity is zero(4)
Using the available theorems, such as Cauchy’s integral theorem and Laurent’s theorem, to simplify the stress function


### 2.1.1. Complex Variable Function

In the absence of body forces, the stresses around a cavity can be solved using the Airy stress functionF as(1)∂4F∂y4+2∂4F∂x2∂y2+∂4F∂x4=0,where x and y denote the horizontal and vertical axes, respectively, in a Cartesian coordinate system, as shown in Figure 1.Figure 1
Thez-plane.The equilibrium can be satisfied ifF meets the following conditions [12]:(2)σx=∂2F∂y2,σy=∂2F∂x2,τxy=−∂2F∂x∂y,where σx and σy are the normal stresses in the x and y directions, respectively, and τxy is the shear stress in the x-y plane.The displacements around the cavity can be expressed by the two harmonic functions,φz and ψz, as follows [34]:(3)2Gux+iuy=κφz−zφ′z¯−ψz¯,where ux and uy are the displacements in the x and y directions, respectively; G is the shear modulus of the soil; z is a complex variable in the z-plane, z=x+iy; the constant κ is related to Poisson’s ratio v by κ=3−4v for the plane strain problems and κ=3−v/1+v for the plane stress problems, respectively; ν is Poisson’s ratio of soil; and φz and ψz are functions with respect to the variable z, which can be determined by giving the displacement boundary conditions around the tunnel cavity. φ′z¯ and ψz¯ are the conjugates of φ′z¯ and ψz¯, respectively.Afterward, the stresses around the cavity can be expressed in the form of complex functions as follows:(4)σx+σy=4Reφ′z,σy−σx+2iτxy=2z¯φ″z+ψ′z,where z¯ is the conjugate of z and Re[.] denotes the real part of a generic complex variable [⋅].


### 2.1.2. Conformal Mapping

A tunnel with an arbitrary cross section (ellipse, circle, oval, and rectangle) in thez-plane is transformed into a unit circular hole in the ζ-plane, as depicted in Figure 2. The employment of the mapping technique z=wζ yieldFigure 2
Conformal mapping of the exterior of an unlined tunnel of any shape onto the exterior of a unit circle.(5)φz=φwζ=φζ,ψz=ψwζ=ψζ.For a 2-dimensional (2D) elastic problem, the displacements and stresses are expressed in terms of two complex variable functions,φ(z) and ψ(z), with z = x + iy and i=−1. By introducing conformal mapping and equation (5), the generated stresses and displacements around the tunnel cavity can be expressed as follows:(6)2Gux+iuy=κφζ−wζw′ζ¯φ′ζ¯−ψζ,¯(7)σx+σy=4Reφ′ζw′ζ,σy−σx+2iτxy=2w′ζwζ¯φ′ζw′ζ′+ψ′ζ,where wζ¯ is the conjugate of the complex function wζ, ζ is a complex variable in the ζ-plane, ζ=ξ+iη=ρeiϑ, ξ and η are variables of the Cartesian coordinates system in the mapped plane, and ρ and ϑ are variables of the polar coordinates system in the mapped plane.From (7), the stress components can be obtained as follows:(8)τxy=Im1w′ζwζ¯φ′ζw′ζ′+ψ′ζ,σx=2Reφ′ζw′ζ−Re1w′ζwζ¯φ′ζw′ζ′+ψ′ζ,σx=2Reφ′ζw′ζ−Re1w′ζwζ¯φ′ζw′ζ′+ψ′ζ,where Im[⋅] denotes the imaginary component of the generic complex variable [⋅].Since (6) is analytic in the mapped ζ-plane, it can be expanded in the Laurent series of σ using ζ=ρeiϑ=ρσ as follows:(9)2Gux+iuy=∑−∞∞Akeinθ=∑−∞∞Anσn=gσ,where σ denotes a point on the unit circle in the ζ-plane. Therefore, the coefficients Ak are determined by means of the recursive relations derived from the given boundary conditions.The displacement components can be obtained as follows:(10)ux=Regσ=12GRe∑−∞∞Anσn,uy=Imgσ=12GIm∑−∞∞Anσn.As shown by Muskhelishvili, the stress boundary condition in the phase plane can be expressed as follows:(11)φζ+wζwζ¯φ′ζ¯+ψζ¯=i∫fx+fyds,where fx and fy are the surface stresses in the x and y directions, respectively, where they act on the elliptical cavity wall.It can be verified [34] that the potentials can be written in the form as follows:(12)φζ=18π1−vFx¯+iFy¯lnζ+Bwζ+φ0ζ,(13)ψζ=−3−4v8π1−vFx¯−iFy¯lnζ+B′+iC′wζ+φ0ζ,where Fx¯+iFy¯ is the resultant force applied to the boundary (which in this case is equal to zero). The variables B, B’ + iC’, and B’ − iC’ can be written as(14)B=σ1+σ24,(15)B′+iC′=−12σ1−σ2e−2iα,where σ1 and σ2 are the principal stresses at infinity and α is the angle between the maximum principal direction and x-direction.It is assumed that the stresses vanished at infinity and on the tunnel boundary. Thus,(16)Fx¯+iFy¯=0,B′+iC′=0.By substituting (16) into (12) and (13),(17)φζ=φ0ζ,ψζ=ψ0ζ,where the functions φ0ζ and ψ0ζ are single-valued and analytic in the ζ-plane, including the point at infinity.According to equations (6), (9), and (17),(18)gσ=∑−∞∞Anσn=3−4vφ0σ−wσw′σ¯φ0′σ¯−ψ0σ¯.By taking the conjugate, then(19)gσ¯=∑−∞∞Anσ−n¯=3−4vφ0σ¯−wσ¯w′σφ0′σ−ψ0σ.Equations19 and 20 are multiplied by dσ/2πiσ−ζ and integrated along the unit circle in the ζ-plane:(20)12πi3−4v∮φ0σσ−ζdσ−12πi∮wσw′σ¯φ0′σ¯σ−ζdσ−12πi∮ψ0σ¯σ−ζdσ=12πi∮gσσ−ζdσ,(21)12πi3−4v∮φ0σ¯σ−ζdσ−12πi∮wσ¯w′σφ0′σσ−ζdσ−12πi∮ψ0σσ−ζdσ=12πi∮gσ¯σ−ζdσ.According to the properties of the Cauchy integral,wσ¯,w′σ,andφ0′σ are analytic functions in the unit circle. Then, wσ¯/w′σφ0′σ is analytic in the unit circle. It can be concluded that the second and third terms on the left-hand side of (17) are zero and that the first term on the left-hand side of (18) is zero. Thus, it can be concluded that(22)φ0ζ=−13−4v12πi∮gσσ−ζdσ.By combining (9) and (23),(23)φ0ζ=13−4v∑−∞0Anζn.A similar process could be performed to findψ0ζ, where An are the only unknown constants in the two complex variable functions.


### 2.1.3. Tunnels with Different Cross Sections

To find a solution for the present problem, we first consider the transformation of the tunnels with arbitrary cross sections, such as ellipses, circles, oval shapes, and squares, in thez-plane into a unit circular hole in the ζ-plane. The transformation function [35] is assumed as(24)z=wς=Rς+∑n=1Nanς−n,ς=ρeiθ,where R is a real number that refers to the cross-section size and an is a general complex coefficient satisfying an<1/n [36]. Inverse mapping is analytical, single-valued, and nonzero in the exterior part of the curve.In many instances, it is assumed that the physical domain possessesp symmetry axes and then yields(25)z=wς=Rς+∑n=1Na1pnς1−pn.The tunnel cavity assigns the prescribed radial displacement along its boundary, and the central position of the curve remains unchanged after deformation. It remains axisymmetric along thex-axis and y-axis. After deformation, the curve can be expressed as(26)Z=wcζ=Rcζ+∑n=1Nbnζ−n.The total displacementuk at the tunnel boundary can be expressed as(27)u=Z−z,=ux+iuy,=Rc−Rζ+∑n=1NRcbn−Ranζ−n.Compared with (10), the coefficient Ak is obtained as follows:(28)A1=2GRc−RAn=0n≥2An=2GRcbk−Rakn≤−1.For a cylindrical cavity of radiusr in isotropic soil, the boundary condition is solved by mapping onto a circle of a unit radius:(29)z=rζ.When undergoing uniform convergenceuε, the deformed tunnel boundary is expressed as follows:(30)Z=r−uεζ.Considering an infinite plane containing an elliptical tunnel cavity with a major axis 2a1 and a minor axis 2b1, the transformation function became(31)z=wζ,=R0ζ+mζ−1,where R0=a1+b1/2andm=a1−b1/a1+b1.Heller et al. [37] provide a mapping function for a rectangular opening of unit width and height, K, using the Schwarz–Christoffel integral:(32)z=Rζ+1−K241ζ−K2241ζ3−K21−K21601ζ5−K21−3K2+K48961ζ7+⋯.Manh et al. [16] propose that, to get a smaller error for the elastic fields at the corner of a rectangular opening, at least 10 terms must be used in the conformal mapping functions. Once an approximate polynomial mapping for the origin and deformed shape of a cavity boundary is known, the elastic solution can be determined using the analytical method proposed in the paper. Generally, tunnel contours comprise complex curves. To generalize these studies, it is reasonable to simplify complicated tunnels as unlined tunnels with elliptical outlines [38].


### 2.1.4. Different Deformation Patterns

It is of practical significance to determine the coefficients of the approximate polynomial mapping function after deformation. Taking the elliptical tunnel as an example, four convergence patterns in Figure3 are assumed to describe the displacement on the boundary of the elliptical tunnel:Figure 3
Boundary conditions of the given displacement.(33)B.C.−1:uc=−u0sinθ+tcos2θ,(34)B.C.−2:uc=−u01+sinθ−1−tcos2θ,(35)B.C.−3:uc=−u01+sinθ−1−t2cos2θ,(36)B.C.−4:uc=−u045+3sinθ−5−2tcos2θ,1≤t≤a1b1.It should be noted that the given convergence patterns are reduced to the boundary conditions given by [19] when t=1 and a1=b1.On the boundary of the elliptical cavity in thez-plane,(37)zc=xc+iyc=acosθ+ibsinθz¯c=xc−iyc=acosθ−ibsinθ⇒cosθ=zc+zc¯2a,sinθ=zc−zc¯2bi.On the boundary of the unit circle in theζ-plane,(38)ζ=eiϑ=σ,ζ¯=e−iϑ=σ−1,(39)zc=Rζc+mζc=Rσ+mσ−1,zc¯=Rζc¯+mζc¯=Rσ−1+mσ.By substituting (35) into (3),(40)cosθ=R1+mσ+σ−12a,sinθ=R1−mσ−σ−12bi,(41)Fx,y=2Gux20+iuy20=2Guccosθ+icosθ.By substituting (31)–(34) and (38) into (39), the displacement can be expressed as a polynomial of σk. The constant Ak is obtained through a comparison with (9). By substituting Ak in (8), the stress components are found.


## 2.1.1. Complex Variable Function

In the absence of body forces, the stresses around a cavity can be solved using the Airy stress functionF as(1)∂4F∂y4+2∂4F∂x2∂y2+∂4F∂x4=0,where x and y denote the horizontal and vertical axes, respectively, in a Cartesian coordinate system, as shown in Figure 1.Figure 1
Thez-plane.The equilibrium can be satisfied ifF meets the following conditions [12]:(2)σx=∂2F∂y2,σy=∂2F∂x2,τxy=−∂2F∂x∂y,where σx and σy are the normal stresses in the x and y directions, respectively, and τxy is the shear stress in the x-y plane.The displacements around the cavity can be expressed by the two harmonic functions,φz and ψz, as follows [34]:(3)2Gux+iuy=κφz−zφ′z¯−ψz¯,where ux and uy are the displacements in the x and y directions, respectively; G is the shear modulus of the soil; z is a complex variable in the z-plane, z=x+iy; the constant κ is related to Poisson’s ratio v by κ=3−4v for the plane strain problems and κ=3−v/1+v for the plane stress problems, respectively; ν is Poisson’s ratio of soil; and φz and ψz are functions with respect to the variable z, which can be determined by giving the displacement boundary conditions around the tunnel cavity. φ′z¯ and ψz¯ are the conjugates of φ′z¯ and ψz¯, respectively.Afterward, the stresses around the cavity can be expressed in the form of complex functions as follows:(4)σx+σy=4Reφ′z,σy−σx+2iτxy=2z¯φ″z+ψ′z,where z¯ is the conjugate of z and Re[.] denotes the real part of a generic complex variable [⋅].


## 2.1.2. Conformal Mapping

A tunnel with an arbitrary cross section (ellipse, circle, oval, and rectangle) in thez-plane is transformed into a unit circular hole in the ζ-plane, as depicted in Figure 2. The employment of the mapping technique z=wζ yieldFigure 2
Conformal mapping of the exterior of an unlined tunnel of any shape onto the exterior of a unit circle.(5)φz=φwζ=φζ,ψz=ψwζ=ψζ.For a 2-dimensional (2D) elastic problem, the displacements and stresses are expressed in terms of two complex variable functions,φ(z) and ψ(z), with z = x + iy and i=−1. By introducing conformal mapping and equation (5), the generated stresses and displacements around the tunnel cavity can be expressed as follows:(6)2Gux+iuy=κφζ−wζw′ζ¯φ′ζ¯−ψζ,¯(7)σx+σy=4Reφ′ζw′ζ,σy−σx+2iτxy=2w′ζwζ¯φ′ζw′ζ′+ψ′ζ,where wζ¯ is the conjugate of the complex function wζ, ζ is a complex variable in the ζ-plane, ζ=ξ+iη=ρeiϑ, ξ and η are variables of the Cartesian coordinates system in the mapped plane, and ρ and ϑ are variables of the polar coordinates system in the mapped plane.From (7), the stress components can be obtained as follows:(8)τxy=Im1w′ζwζ¯φ′ζw′ζ′+ψ′ζ,σx=2Reφ′ζw′ζ−Re1w′ζwζ¯φ′ζw′ζ′+ψ′ζ,σx=2Reφ′ζw′ζ−Re1w′ζwζ¯φ′ζw′ζ′+ψ′ζ,where Im[⋅] denotes the imaginary component of the generic complex variable [⋅].Since (6) is analytic in the mapped ζ-plane, it can be expanded in the Laurent series of σ using ζ=ρeiϑ=ρσ as follows:(9)2Gux+iuy=∑−∞∞Akeinθ=∑−∞∞Anσn=gσ,where σ denotes a point on the unit circle in the ζ-plane. Therefore, the coefficients Ak are determined by means of the recursive relations derived from the given boundary conditions.The displacement components can be obtained as follows:(10)ux=Regσ=12GRe∑−∞∞Anσn,uy=Imgσ=12GIm∑−∞∞Anσn.As shown by Muskhelishvili, the stress boundary condition in the phase plane can be expressed as follows:(11)φζ+wζwζ¯φ′ζ¯+ψζ¯=i∫fx+fyds,where fx and fy are the surface stresses in the x and y directions, respectively, where they act on the elliptical cavity wall.It can be verified [34] that the potentials can be written in the form as follows:(12)φζ=18π1−vFx¯+iFy¯lnζ+Bwζ+φ0ζ,(13)ψζ=−3−4v8π1−vFx¯−iFy¯lnζ+B′+iC′wζ+φ0ζ,where Fx¯+iFy¯ is the resultant force applied to the boundary (which in this case is equal to zero). The variables B, B’ + iC’, and B’ − iC’ can be written as(14)B=σ1+σ24,(15)B′+iC′=−12σ1−σ2e−2iα,where σ1 and σ2 are the principal stresses at infinity and α is the angle between the maximum principal direction and x-direction.It is assumed that the stresses vanished at infinity and on the tunnel boundary. Thus,(16)Fx¯+iFy¯=0,B′+iC′=0.By substituting (16) into (12) and (13),(17)φζ=φ0ζ,ψζ=ψ0ζ,where the functions φ0ζ and ψ0ζ are single-valued and analytic in the ζ-plane, including the point at infinity.According to equations (6), (9), and (17),(18)gσ=∑−∞∞Anσn=3−4vφ0σ−wσw′σ¯φ0′σ¯−ψ0σ¯.By taking the conjugate, then(19)gσ¯=∑−∞∞Anσ−n¯=3−4vφ0σ¯−wσ¯w′σφ0′σ−ψ0σ.Equations19 and 20 are multiplied by dσ/2πiσ−ζ and integrated along the unit circle in the ζ-plane:(20)12πi3−4v∮φ0σσ−ζdσ−12πi∮wσw′σ¯φ0′σ¯σ−ζdσ−12πi∮ψ0σ¯σ−ζdσ=12πi∮gσσ−ζdσ,(21)12πi3−4v∮φ0σ¯σ−ζdσ−12πi∮wσ¯w′σφ0′σσ−ζdσ−12πi∮ψ0σσ−ζdσ=12πi∮gσ¯σ−ζdσ.According to the properties of the Cauchy integral,wσ¯,w′σ,andφ0′σ are analytic functions in the unit circle. Then, wσ¯/w′σφ0′σ is analytic in the unit circle. It can be concluded that the second and third terms on the left-hand side of (17) are zero and that the first term on the left-hand side of (18) is zero. Thus, it can be concluded that(22)φ0ζ=−13−4v12πi∮gσσ−ζdσ.By combining (9) and (23),(23)φ0ζ=13−4v∑−∞0Anζn.A similar process could be performed to findψ0ζ, where An are the only unknown constants in the two complex variable functions.


## 2.1.3. Tunnels with Different Cross Sections

To find a solution for the present problem, we first consider the transformation of the tunnels with arbitrary cross sections, such as ellipses, circles, oval shapes, and squares, in thez-plane into a unit circular hole in the ζ-plane. The transformation function [35] is assumed as(24)z=wς=Rς+∑n=1Nanς−n,ς=ρeiθ,where R is a real number that refers to the cross-section size and an is a general complex coefficient satisfying an<1/n [36]. Inverse mapping is analytical, single-valued, and nonzero in the exterior part of the curve.In many instances, it is assumed that the physical domain possessesp symmetry axes and then yields(25)z=wς=Rς+∑n=1Na1pnς1−pn.The tunnel cavity assigns the prescribed radial displacement along its boundary, and the central position of the curve remains unchanged after deformation. It remains axisymmetric along thex-axis and y-axis. After deformation, the curve can be expressed as(26)Z=wcζ=Rcζ+∑n=1Nbnζ−n.The total displacementuk at the tunnel boundary can be expressed as(27)u=Z−z,=ux+iuy,=Rc−Rζ+∑n=1NRcbn−Ranζ−n.Compared with (10), the coefficient Ak is obtained as follows:(28)A1=2GRc−RAn=0n≥2An=2GRcbk−Rakn≤−1.For a cylindrical cavity of radiusr in isotropic soil, the boundary condition is solved by mapping onto a circle of a unit radius:(29)z=rζ.When undergoing uniform convergenceuε, the deformed tunnel boundary is expressed as follows:(30)Z=r−uεζ.Considering an infinite plane containing an elliptical tunnel cavity with a major axis 2a1 and a minor axis 2b1, the transformation function became(31)z=wζ,=R0ζ+mζ−1,where R0=a1+b1/2andm=a1−b1/a1+b1.Heller et al. [37] provide a mapping function for a rectangular opening of unit width and height, K, using the Schwarz–Christoffel integral:(32)z=Rζ+1−K241ζ−K2241ζ3−K21−K21601ζ5−K21−3K2+K48961ζ7+⋯.Manh et al. [16] propose that, to get a smaller error for the elastic fields at the corner of a rectangular opening, at least 10 terms must be used in the conformal mapping functions. Once an approximate polynomial mapping for the origin and deformed shape of a cavity boundary is known, the elastic solution can be determined using the analytical method proposed in the paper. Generally, tunnel contours comprise complex curves. To generalize these studies, it is reasonable to simplify complicated tunnels as unlined tunnels with elliptical outlines [38].


## 2.1.4. Different Deformation Patterns

It is of practical significance to determine the coefficients of the approximate polynomial mapping function after deformation. Taking the elliptical tunnel as an example, four convergence patterns in Figure3 are assumed to describe the displacement on the boundary of the elliptical tunnel:Figure 3
Boundary conditions of the given displacement.(33)B.C.−1:uc=−u0sinθ+tcos2θ,(34)B.C.−2:uc=−u01+sinθ−1−tcos2θ,(35)B.C.−3:uc=−u01+sinθ−1−t2cos2θ,(36)B.C.−4:uc=−u045+3sinθ−5−2tcos2θ,1≤t≤a1b1.It should be noted that the given convergence patterns are reduced to the boundary conditions given by [19] when t=1 and a1=b1.On the boundary of the elliptical cavity in thez-plane,(37)zc=xc+iyc=acosθ+ibsinθz¯c=xc−iyc=acosθ−ibsinθ⇒cosθ=zc+zc¯2a,sinθ=zc−zc¯2bi.On the boundary of the unit circle in theζ-plane,(38)ζ=eiϑ=σ,ζ¯=e−iϑ=σ−1,(39)zc=Rζc+mζc=Rσ+mσ−1,zc¯=Rζc¯+mζc¯=Rσ−1+mσ.By substituting (35) into (3),(40)cosθ=R1+mσ+σ−12a,sinθ=R1−mσ−σ−12bi,(41)Fx,y=2Gux20+iuy20=2Guccosθ+icosθ.By substituting (31)–(34) and (38) into (39), the displacement can be expressed as a polynomial of σk. The constant Ak is obtained through a comparison with (9). By substituting Ak in (8), the stress components are found.


## 2.2. Application to an Elliptical Tunnel in a Full Plane

Considering that an infinite plane contains an elliptical tunnel cavity with a major axis 2a1 and a minor axis 2b1, as illustrated in Figure 1, a uniform radial displacement (B.C.1, t=1) is assumed as the boundary condition for the elliptical tunnel cavity. The initial elliptical tunnel cavity is converged without altering the ratio of the semimajor axis to the semiminor axis of the elliptical tunnel cavity.(42)a1a2=b1b2,=ρ1ρ2,=1k,where a2 and b2 correspond to the current semimajor and semiminor axes of the elliptical tunnel cavity, respectively; ρ1,ρ2 denote the angular length of the elliptical tunnel at any point before and after the deformation in the polar coordinate system, respectively; and k is the convergence ratio.At the boundary of the elliptical tunnel cavity,(43)ux0=−1−k,x=−1−kz+z¯2=−1−kRm+1σ+m+1σ−12,uy0=−1−k,y=−1−kz−z¯2=−1−kR1−mσ−1−mσ−12.By comparing (23) and (25), there are only two Fourier coefficients for the complex variable function:(44)A1=2G1−kR,A−1=2G1−kRm,Ak=0k>1,k<−1.


### 2.2.1. Comparison of the Solutions for the Elliptical and Circular Tunnels

The ratio of the semimajor and semiminor axes of the initial elliptical cavity (a/b) is 5/4, and the soil Poisson’s ratio equals 0.5. The elastic solution of the elliptical tunnel cavity and the circumscribed circular tunnel cavity expansion shown in Figure 4 are obtained using the analytical method. The orthoradial displacement and shearing stress caused by the radial displacement of the circumscribed circular tunnel cavity and elliptical tunnel cavity along the semimajor and semiminor axis directions of the elliptical tunnel cavity are zero.Figure 4
Current elliptical tunnel cavity and circumscribed circular tunnel cavity before and after the radial displacement at the boundary of the original tunnel cavity.Figure5 shows the radial displacement variation of the soil in relation to the distance from the center of the cavities, normalized by the radius of the circumscribed circular tunnel cavity. The analytical results of the radial displacement along the major and minor axes of the elliptical tunnel cavity are compared with the results of the circumscribed circular tunnel cavity.Figure 5
Distribution of the radial displacement of the soil due to the radial displacement at the boundaries of the cavities.It can be seen that the radial displacement of the soil around the elliptical tunnel cavity along the major and minor axes and circumscribed circular tunnel cavity decreases with the increase in the distance from the cavity’s center and that it eventually tends to zero. The radial displacement of the soil along the major axis of the elliptical tunnel cavity is equal to that of the circumscribed circular tunnel cavity.Figure6 shows the radial and orthoradial stresses of the soil, which are caused by the radial displacement of the elliptical tunnel cavity and circumscribed circular tunnel cavity in relation to the distance from the center of the cavities. The magnitude of the radial stress of the soil along the semimajor axis of the elliptical tunnel cavity is larger than the radial stress of the soil along the radial direction of the circumscribed circular. The radial stresses along the semiminor axis of the elliptical tunnel cavity are minimum. The magnitude of the soil orthoradial stress induced by the radial displacement of the circumscribed circular tunnel cavity is larger than the soil orthoradial stress caused by the elliptical tunnel cavity. As the distance from the cavity center increased, the soil radial and orthoradial stresses around the elliptical tunnel cavity and circumscribed circular tunnel cavity tend to zero.Figure 6
Distribution of the stress of the soil elements due to the radial displacement at the boundaries of the cavities.Whena=4m,G=2000kPa,andk=0.92 are considered, the magnitudes of the soil stress and displacement around the cavities are very small σρ<1kPa,σθ<1kPa,τρθ<1kPa,uρ<0.010mm, at ∼50 m from the cavity center. That is, the influence radius of the convergence of the elliptical cavity is ∼20 times that of the semimajor axis.


### 2.2.2. Elastic Solutions of the Soil around an Elliptical Tunnel Cavity Using the Analytical and Finite Element Method Methods

To validate the accuracy of the results, the solution is compared with a finite element method (FEM) calculation using the ABAQUS 2D software. The FEM model size is1000×1000m, and the boundary elements were infinite elements. Since the whole model is symmetric about the origin point and the x, y axes, one-quarter of the whole model is selected for comparison. The soil’s stress and displacement around the elliptical ellipse are compared and analyzed. Figures 7 and 8 show the contour of the normalized stresses and displacements of the soil from the analytical solution with the FEM solution. The selected parameters of the elliptical tunnel cavity and soil are as follows:(45)a1H=547,k=0.92,G=2000kPa,v=0.4.where a is the semimajor axis, k is the convergence ratio, G is the shear modulus, and v is the soil Poisson’s ratio.Figure 7
Contours of the normalized stresses and displacements in the case of (a)/(b) = 2 as derived by the theoretical method and ABAQUS 2D. (a) Shear stress of the soil. (b) Soil stress in thex-direction. (c) Soil stress in the y-direction. (d) Soil displacement in the x-direction. (e) Soil displacement in the y-direction.
(a)(b)(c)(d)(e)Figure 8
Contours of the normalized stresses and displacements in the case of the elliptical tunnel with different ellipticity values. (a) Shearing stress of soil. (b) Soil stress in thex-direction. (c) Soil displacement in the x-direction. (d) Soil displacement in the y-direction. (e) Soil stress in the y-direction.
(a)(b)(c)(d)(e)It can be seen from Figures7(a)–7(e) that the soil shearing stress in the directions of the x-axis and the y-axis is zero. There are some differences between the values of the vertical stresses and displacements because the boundary condition of the ABAQUS model is finite and fixed at the bottom. The soil stress and displacement calculated using the complex function method and FEM are approximately equal, which verifies the correctness of the theoretical method in this paper.


### 2.2.3. Elastic Solutions of the Soil around the Elliptical Tunnel Cavity with Different Ellipticity Values

Figures8(a)–8(e) show the soil stress and displacement caused by the convergence of the elliptical cavities with different ellipticity values. From the figures, it can be seen that the ellipticity of the cavity decreased, the soil stress in the x-axis direction of the soil increased, and that the soil stress and displacement in the y-axis direction decreases. Also, the soil displacement in the x-axis direction can be seen.


## 2.2.1. Comparison of the Solutions for the Elliptical and Circular Tunnels

The ratio of the semimajor and semiminor axes of the initial elliptical cavity (a/b) is 5/4, and the soil Poisson’s ratio equals 0.5. The elastic solution of the elliptical tunnel cavity and the circumscribed circular tunnel cavity expansion shown in Figure 4 are obtained using the analytical method. The orthoradial displacement and shearing stress caused by the radial displacement of the circumscribed circular tunnel cavity and elliptical tunnel cavity along the semimajor and semiminor axis directions of the elliptical tunnel cavity are zero.Figure 4
Current elliptical tunnel cavity and circumscribed circular tunnel cavity before and after the radial displacement at the boundary of the original tunnel cavity.Figure5 shows the radial displacement variation of the soil in relation to the distance from the center of the cavities, normalized by the radius of the circumscribed circular tunnel cavity. The analytical results of the radial displacement along the major and minor axes of the elliptical tunnel cavity are compared with the results of the circumscribed circular tunnel cavity.Figure 5
Distribution of the radial displacement of the soil due to the radial displacement at the boundaries of the cavities.It can be seen that the radial displacement of the soil around the elliptical tunnel cavity along the major and minor axes and circumscribed circular tunnel cavity decreases with the increase in the distance from the cavity’s center and that it eventually tends to zero. The radial displacement of the soil along the major axis of the elliptical tunnel cavity is equal to that of the circumscribed circular tunnel cavity.Figure6 shows the radial and orthoradial stresses of the soil, which are caused by the radial displacement of the elliptical tunnel cavity and circumscribed circular tunnel cavity in relation to the distance from the center of the cavities. The magnitude of the radial stress of the soil along the semimajor axis of the elliptical tunnel cavity is larger than the radial stress of the soil along the radial direction of the circumscribed circular. The radial stresses along the semiminor axis of the elliptical tunnel cavity are minimum. The magnitude of the soil orthoradial stress induced by the radial displacement of the circumscribed circular tunnel cavity is larger than the soil orthoradial stress caused by the elliptical tunnel cavity. As the distance from the cavity center increased, the soil radial and orthoradial stresses around the elliptical tunnel cavity and circumscribed circular tunnel cavity tend to zero.Figure 6
Distribution of the stress of the soil elements due to the radial displacement at the boundaries of the cavities.Whena=4m,G=2000kPa,andk=0.92 are considered, the magnitudes of the soil stress and displacement around the cavities are very small σρ<1kPa,σθ<1kPa,τρθ<1kPa,uρ<0.010mm, at ∼50 m from the cavity center. That is, the influence radius of the convergence of the elliptical cavity is ∼20 times that of the semimajor axis.


## 2.2.2. Elastic Solutions of the Soil around an Elliptical Tunnel Cavity Using the Analytical and Finite Element Method Methods

To validate the accuracy of the results, the solution is compared with a finite element method (FEM) calculation using the ABAQUS 2D software. The FEM model size is1000×1000m, and the boundary elements were infinite elements. Since the whole model is symmetric about the origin point and the x, y axes, one-quarter of the whole model is selected for comparison. The soil’s stress and displacement around the elliptical ellipse are compared and analyzed. Figures 7 and 8 show the contour of the normalized stresses and displacements of the soil from the analytical solution with the FEM solution. The selected parameters of the elliptical tunnel cavity and soil are as follows:(45)a1H=547,k=0.92,G=2000kPa,v=0.4.where a is the semimajor axis, k is the convergence ratio, G is the shear modulus, and v is the soil Poisson’s ratio.Figure 7
Contours of the normalized stresses and displacements in the case of (a)/(b) = 2 as derived by the theoretical method and ABAQUS 2D. (a) Shear stress of the soil. (b) Soil stress in thex-direction. (c) Soil stress in the y-direction. (d) Soil displacement in the x-direction. (e) Soil displacement in the y-direction.
(a)(b)(c)(d)(e)Figure 8
Contours of the normalized stresses and displacements in the case of the elliptical tunnel with different ellipticity values. (a) Shearing stress of soil. (b) Soil stress in thex-direction. (c) Soil displacement in the x-direction. (d) Soil displacement in the y-direction. (e) Soil stress in the y-direction.
(a)(b)(c)(d)(e)It can be seen from Figures7(a)–7(e) that the soil shearing stress in the directions of the x-axis and the y-axis is zero. There are some differences between the values of the vertical stresses and displacements because the boundary condition of the ABAQUS model is finite and fixed at the bottom. The soil stress and displacement calculated using the complex function method and FEM are approximately equal, which verifies the correctness of the theoretical method in this paper.


## 2.2.3. Elastic Solutions of the Soil around the Elliptical Tunnel Cavity with Different Ellipticity Values

Figures8(a)–8(e) show the soil stress and displacement caused by the convergence of the elliptical cavities with different ellipticity values. From the figures, it can be seen that the ellipticity of the cavity decreased, the soil stress in the x-axis direction of the soil increased, and that the soil stress and displacement in the y-axis direction decreases. Also, the soil displacement in the x-axis direction can be seen.


## 3. Elastic Solutions in a Half-Plane

### 3.1. Virtual Image Technique

So far, the displacement fields are considered in an infinite plane. The ground displacement induced by the deformation of the tunnel cavity with the arbitrary cross section in the half-plane can be obtained using the virtual image technique.Figure9 shows the diagram of the virtual image technique used in this paper. Supposing that y = 0 is the surface, there are two convergence elliptical cavities at the points O1 (0, −h) and O2 (0, h). Then, at the surface,(46)uy|y=0=0;ux|y=0=2uxx,h,σy|y=0=2σyx,h;τxy|y=0=0.Figure 9
Diagram of the virtual image technique.The boundary of the half-plane is considered to be stress-free. The normal stresses at the surfacey=0 are made to be equal to zero, meaning that it is necessary to apply a reverse force to balance the normal stress at the surface induced by the two convergence elliptical tunnel cavities. This problem turns into a Boussinesq problem for the half planey≤0 with the following boundary conditions:(47)y=0:σy1=−2σyx,0,τxyx,0=0.In the case of a circular tunnel, the ground displacement induced by the reverse vertical stress applied to the ground surface can be solved using the Fourier formula [23]. When the section shape of the tunnel is elliptical, the vertical stress expression is complex, and the Fourier transform is no longer applicable. In this paper, the numerical solution of the surface displacement can be derived using the formula proposed by [39].Reference [39] uses the complex variable function and virtual image technique to obtain the displacement caused by the vertical and horizontal concentrated forces at any point in the half plane. When a unit vertical concentrated force applies to a point m in the half plane, as illustrated in Figure 10, the stress function becomes(48)φ1z=iP8π1−vlogz−ihz+ih−iP2πlogz−ih+P4π1−vhz−ih,ψ1z=iP8π1−v3−4vlogz−ihz+ih+ihz−ih+ihz+ih,−iP2πlogz−ih−P1−2v4π1−vihz−ih+P4π1−vhzz−ih2,(49)2Gux+iuy=κφ1z−zφ1′z¯−ψ1z.¯Figure 10
Schematic diagram of the vertical and horizontal concentrated forces applied to the half-plane.
(a)(b)Similarly, the stress function is obtained when a unit horizontal concentrated force is applied to a pointm in the half plane, as seen in Figure 10(b). Supposing that h=0 in (44) and (43), the surface displacement caused by the unit force applied to the surface can be obtained.Figure11 shows a schematic diagram of solving the vertical displacement method. Due to the concentrated force P applied to the boundary, the subsidence of the point K to the boundary (the distance from the origin is r) relative to the subsidence of the reference point B (the distance from the origin is s) is(50)η=2P1−v2πElnsr.Figure 11
Schematic diagram of solving the vertical displacement method.The relative horizontal displacement is(51)ux=−1+vπE3−4v41−vlnr2s2+1−2v221−vlnrs,where r is the distance from the position of the concentrated force P to the point K and s is the distance between the position of the concentrated force P and the reference point B. The method adopted in this paper is mainly to divide the applied uneven vertical load into n parts. As shown in Figure 10, the vertical load applied to the boundary was symmetrical about the y-axis, and the magnitude of the vertical load at infinity was zero. The reference point B is the start point of the 1st load, and point K is the start point of the j-th load. The settlement of point K could be expressed as follows:(52)ηK=∑i=1n2qiΔsπElni−1Δsi−1Δs−rBKrBK≠i−1Δs,rBK=j−1Δs.The relative settlement of pointK to the point B could be expressed as follows:(53)ηK=∑i=1n2qiΔsπEIni−1i−ji≠j.The farther the distance from the reference pointB to the origin point and the finer the load element division, the more accurate the results.


### 3.2. Vertical and Horizontal Displacements Induced by the Elliptical Tunnel

Figure12 shows that when the buried depth of the tunnel decreased, the ground displacement caused by the elliptical tunnel decreased. The vertical displacement of any point on the ground in Figure 13(a) is relative to the vertical displacement of point x=−100. Since the horizontal stress applied to the ground is symmetrical about the y-axis, the horizontal displacement of the ground should be symmetrical with respect to the origin point, and the horizontal displacement of the origin point is zero, so the relative horizontal displacement of any point subtracts the horizontal displacement of the origin point can obtain the absolute value of the horizontal displacement of the point, as shown in Figure 13(b). It can be seen from the above two figures that when the depth of the elliptical tunnel decreased, the surface displacement caused by the contraction of the elliptical tunnel gradually decreases. When Poisson’s ratio decreases, the relative vertical displacement and horizontal displacement of the ground caused by the elliptical tunnel increases.Figure 12
Vertical displacement of ground whena/H changes.Figure 13
Influence of Poisson’s ratio on the horizontal and vertical displacements of ground induced by the elliptical tunnel cavity’s deformation. (a) Vertical displacement of ground. (b) Horizontal displacement of ground.
(a)(b)


## 3.1. Virtual Image Technique

So far, the displacement fields are considered in an infinite plane. The ground displacement induced by the deformation of the tunnel cavity with the arbitrary cross section in the half-plane can be obtained using the virtual image technique.Figure9 shows the diagram of the virtual image technique used in this paper. Supposing that y = 0 is the surface, there are two convergence elliptical cavities at the points O1 (0, −h) and O2 (0, h). Then, at the surface,(46)uy|y=0=0;ux|y=0=2uxx,h,σy|y=0=2σyx,h;τxy|y=0=0.Figure 9
Diagram of the virtual image technique.The boundary of the half-plane is considered to be stress-free. The normal stresses at the surfacey=0 are made to be equal to zero, meaning that it is necessary to apply a reverse force to balance the normal stress at the surface induced by the two convergence elliptical tunnel cavities. This problem turns into a Boussinesq problem for the half planey≤0 with the following boundary conditions:(47)y=0:σy1=−2σyx,0,τxyx,0=0.In the case of a circular tunnel, the ground displacement induced by the reverse vertical stress applied to the ground surface can be solved using the Fourier formula [23]. When the section shape of the tunnel is elliptical, the vertical stress expression is complex, and the Fourier transform is no longer applicable. In this paper, the numerical solution of the surface displacement can be derived using the formula proposed by [39].Reference [39] uses the complex variable function and virtual image technique to obtain the displacement caused by the vertical and horizontal concentrated forces at any point in the half plane. When a unit vertical concentrated force applies to a point m in the half plane, as illustrated in Figure 10, the stress function becomes(48)φ1z=iP8π1−vlogz−ihz+ih−iP2πlogz−ih+P4π1−vhz−ih,ψ1z=iP8π1−v3−4vlogz−ihz+ih+ihz−ih+ihz+ih,−iP2πlogz−ih−P1−2v4π1−vihz−ih+P4π1−vhzz−ih2,(49)2Gux+iuy=κφ1z−zφ1′z¯−ψ1z.¯Figure 10
Schematic diagram of the vertical and horizontal concentrated forces applied to the half-plane.
(a)(b)Similarly, the stress function is obtained when a unit horizontal concentrated force is applied to a pointm in the half plane, as seen in Figure 10(b). Supposing that h=0 in (44) and (43), the surface displacement caused by the unit force applied to the surface can be obtained.Figure11 shows a schematic diagram of solving the vertical displacement method. Due to the concentrated force P applied to the boundary, the subsidence of the point K to the boundary (the distance from the origin is r) relative to the subsidence of the reference point B (the distance from the origin is s) is(50)η=2P1−v2πElnsr.Figure 11
Schematic diagram of solving the vertical displacement method.The relative horizontal displacement is(51)ux=−1+vπE3−4v41−vlnr2s2+1−2v221−vlnrs,where r is the distance from the position of the concentrated force P to the point K and s is the distance between the position of the concentrated force P and the reference point B. The method adopted in this paper is mainly to divide the applied uneven vertical load into n parts. As shown in Figure 10, the vertical load applied to the boundary was symmetrical about the y-axis, and the magnitude of the vertical load at infinity was zero. The reference point B is the start point of the 1st load, and point K is the start point of the j-th load. The settlement of point K could be expressed as follows:(52)ηK=∑i=1n2qiΔsπElni−1Δsi−1Δs−rBKrBK≠i−1Δs,rBK=j−1Δs.The relative settlement of pointK to the point B could be expressed as follows:(53)ηK=∑i=1n2qiΔsπEIni−1i−ji≠j.The farther the distance from the reference pointB to the origin point and the finer the load element division, the more accurate the results.


## 3.2. Vertical and Horizontal Displacements Induced by the Elliptical Tunnel

Figure12 shows that when the buried depth of the tunnel decreased, the ground displacement caused by the elliptical tunnel decreased. The vertical displacement of any point on the ground in Figure 13(a) is relative to the vertical displacement of point x=−100. Since the horizontal stress applied to the ground is symmetrical about the y-axis, the horizontal displacement of the ground should be symmetrical with respect to the origin point, and the horizontal displacement of the origin point is zero, so the relative horizontal displacement of any point subtracts the horizontal displacement of the origin point can obtain the absolute value of the horizontal displacement of the point, as shown in Figure 13(b). It can be seen from the above two figures that when the depth of the elliptical tunnel decreased, the surface displacement caused by the contraction of the elliptical tunnel gradually decreases. When Poisson’s ratio decreases, the relative vertical displacement and horizontal displacement of the ground caused by the elliptical tunnel increases.Figure 12
Vertical displacement of ground whena/H changes.Figure 13
Influence of Poisson’s ratio on the horizontal and vertical displacements of ground induced by the elliptical tunnel cavity’s deformation. (a) Vertical displacement of ground. (b) Horizontal displacement of ground.
(a)(b)


## 4. Conclusions

The paper proposes an analytical method for modeling ground displacements for the tunnels with arbitrary cross sections in clay. The following main conclusions are drawn:(1)
Once an approximate polynomial mapping for the original and deformed shapes of a cavity boundary is obtained, the elastic solution in the full plane can be determined using the proposed analytical method in this paper.(2)
The magnitude of the radial stress of the soil along the semimajor axis of the elliptical tunnel cavity is larger than the radial stress of the soil along the radial direction of the circumscribed circular. The magnitude of the orthoradial stress of the soil induced by the radial displacement of the circumscribed circular tunnel cavity is larger than the orthoradial stresses of the soil caused by the elliptical tunnel cavity. Also, the influence radius of the convergence of the elliptical cavity is 20 times that of the semimajor axis in the full plane.(3)
A good agreement of elastic solutions is found between the analytical solutions and FEM results in the full plane for the elliptical tunnel. As the distance from the cavity center increases, the soil radial and orthoradial stresses around the elliptical tunnel cavity decrease.(4)
The ellipticity of the tunnel cavity decreases, and the soil stress in thex-axis direction of the soil increases. Also, the soil displacement in the x-axis direction and the soil stress and displacement in the y-axis direction decrease.(5)
The surface displacement in the half-plane can be obtained using the virtual image technique. When the depth of the elliptical tunnel decreases, the surface displacement caused by the contraction of the elliptical tunnel gradually decreases. When Poisson’s ratio of the soil decreases, the relative vertical and horizontal displacements of the ground caused by the elliptical tunnel increase.The solution is under the assumption of elasticity and the certain deformation at the tunnel boundary, and the surcharge loadings and internal forces are not considered in the derivation. The solution is only valid for the shallow tunnels excavated in clay.



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*Source: 9916159-2021-07-08.xml*