Analytical Method to Interpret Displacement in Elastic Anisotropic Soil due to a Tunnel Cavity with an Arbitrary Cross Section

Power China Huadong Engineering Corporation, Zhejiang, Hangzhou 310014, China Engineering Research Center of Smart Rail Transportation of Zhejiang Province, Zhejiang, Hangzhou 310014, China China Railway 11th Bureau Group Co., Ltd., Wuhan 430061, China Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China Hangzhou Metro Group Co., Ltd., Zhejiang, Hangzhou 310003, China


Introduction
Due to the complexity of metro projects, many metro shields use varying cross-sectional shapes for the shield, such as rectangular [1], quasi-rectangular [2,3], elliptical, horseshoe-shaped, and double-O-tube [4]. Research on tunnels with arbitrary cross sections can guide engineers in predicting ground displacement and underground deformation.
Currently, numerous research efforts have focused on ground displacement caused by circular shield tunnels. In this context, four primary methods are employed-empirical formula, analytical, model test, and numerical methods. Although numerical methods can easily identify the elastic solution for an underground excavation, analytical methods provide important information.
Carranzatorres and Fairhurst [5] stated that closed-form results can help engineers assess the general accuracy of numerical analysis and provide valuable means for obtaining insights into the general nature of a solution. Elasticity problems can be solved by combining the complex variable method with numerous theorems arising from analytical functions, such as Cauchy's integral theorem, Laurent's theorem, and theorems on conformal mapping [6,7]. ese theories are widely used for the first fundamental problem of deep tunnels. Bobet [8] developed analytical solutions for a lined deep tunnel with a circular cross section in transversely isotropic rock using the Lkehnitskii formalism [9]. Zhang and Sun [10] derived an analytical solution for the radial displacement of an unlined deep tunnel with an arbitrary cross section in transversely isotropic rock using Kolosov-Muskhelishvili complex potentials [11]. Exadaktylos [12] and Manh [13] proposed closed-form solutions for the stress and displacement around an unlined deep tunnel with an arbitrary cross section in elastic isotropic and anisotropic ground, respectively. Exadaktylos [12] applied the method of Kolosov-Muskhelishvili complex potentials, whereas Manh [13] used Green's theory [14][15][16].
Analytical solutions have been reported for displacement-prescribed problems (also known as the second fundamental problem) in the context of unlined shallow tunnels in elastic soil. Moreover, analytical solutions have been developed for ground movements induced by tunnels with a circular cross section in clay by using the complex variable method. Sagaseta [17] considered tunnel excavation as the radial convergence of a point to the tunnel axis in an elastic half plane. Verruijt [18] extended the work of Sagaseta for arbitrary Poisson ratios and proposed an elliptical deformation mode for the tunnel boundary. Sagaseta found that the deformation modes of the tunnel boundary significantly influence the distribution of stress and displacement at the surface and in the internal soil. Park [19,20] proposed four deformation patterns for tunnel boundaries and established the analytical displacement of the soil due to deep and shallow buried tunnels in clay. Pinto [21] discussed vertical displacement of a circular tunnel undergoing uniform radial deformation or exhibiting an elliptical deformation mode. e elliptical deformation mode was verified by comparing the analytical results with the measured results. Zymnis [22,23] extended the analytical solutions proposed by Pinto to account for the cross-anisotropic stiffness properties of the soil. ese studies show that the ground displacement caused by excavation of a circular tunnel in clay can be accurately predicted if appropriate soil parameters and tunnel deformation patterns are considered.
Using finite element analysis, Simpson et al. [24] confirmed that anisotropy has significant effects on both the magnitude of the surface settlement and shape of the surface settlement trough, whereas nonlinear behavior has only subtle effects. Lee and Rowe [25] showed that the elastic cross-anisotropy of soil has a significant effect on the computed settlements above the tunnel.
Relatively few studies have focused on the displacementprescribed problem for a tunnel with an arbitrary cross section in anisotropic soil. In this paper, it is assumed that the tunnel is so long and the plane-strain condition is employed. Full-plane elastic solutions are proposed for an unlined tunnel with an arbitrary cross section subjected to certain deformation modes. Several typical tunnel shapes are used to illustrate the proposed method. To verify the accuracy of this method, results for an elliptical unlined tunnel in transversely isotropic soil with stiffness parameters that correspond to isotropic conditions are compared with that for isotropic soil. e elastic solutions derived for isotropic soil and London clay are also compared to analyze the effect of anisotropic stiffness. Moreover, analytical solutions of displacements in half plane are obtained by using a virtual image technique. e effect of stiffness parameters, n and m, on the distribution of the predicted displacement for an unlined elliptical tunnel is also analyzed.

General Equations for Anisotropic Soil.
For plane-strain problems, the constitutive relation can be given as where the β ij coefficients can be expressed as material stiffness parameters E 1 , E 2 , v 1 , v 2 , G 2 , and α 0 by Zymnis [22], for example, where α 0 is dip angle of plane with isotropic properties; E 1 , E 2 are Young's moduli of the soil in a direction parallel and normal to the isotropic plane, respectively; v 1 is Poisson's ratio in the plane of isotropy; v 2 is Poisson's ratio in the y ′ direction due to strain in the x ′ direction; and G vh is the shear modulus for strain in the y ′ direction.
For α 0 � 0, e following stiffness parameters are commonly used to measure the anisotropy of soil: Chatzigiannelis and Whittle [21] conducted laboratory tests on elastic anisotropic parameters for various types of soil. eir results are summarized in Table 1.
e elastic parameters are further constrained by thermodynamic considerations [22], such as n hh + 2n hv n vh ≤ 1.

Compatibility Equation for Transverse Isotropy.
e equation for equilibrium under plane-strain conditions is given as follows: In the absence of body forces, the stresses can be derived from the Airy stress function F as follows: Substituting equations (1) and (6) into equation (5), we obtain e general solution can be expressed in terms of two analytic functions F 1 , F 2 and their conjugates: where the overscript "− " denotes the complex conjugate and z k � x + λ k y, k � 1, 2 are arguments in the z k plane. is plane is obtained from the affine transformation as follows: where λ k are the material eigenvalues. By introducing equation (9) into equation (8), we can obtain the characteristic equation of λ k : (11) where the roots of the equation are conjugate complex numbers, λ k , λ k , k � 1, 2.
New generalized complex variable functions are introduced as follows: where Φ k (z k ) can be assumed in the form of an infinite power series. From equations (7) and (12), the stress components are related to the variable functions as follows: Additionally, the cavity is assumed to be free of forces along its surface. e displacement components are determined by integrating the strain components with

Closed-Form Solution Obtained by Conformal Mapping
To find solutions for this problem, we first consider the transformation of tunnels with an arbitrary cross section, such as an ellipse, circle, or square, in the z plane to a unit circular hole in the ζ plane, as shown in Figure 1. e transformation function can be assumed as where R is a real number related to the size of the tunnel and a n are generally complex coefficients that satisfy |a n | < 1/n [26]. e inverse mapping function w − 1 (z) is analytic, single valued, and nonzero outside the tunnel boundary. In many cases, it can be assumed that the physical domain possesses p symmetry axes, which yields Table 1: Transverse isotropic parameters of different types of soil as reported in the literature [21].
Assuming that the central position of the tunnel boundary remains unchanged after deformation, the boundary remains axisymmetric along the x and y axes. After deformation, the curve can be expressed by Because the soil is transversely isotropic, the boundary of the tunnel in the original plane (z plane) is affine transformed to cutouts in the z k planes: e exterior region of an unlined tunnel in the z k plane can be mapped to the exterior region of a unit circle in the ζ k plane. e conformal mapping functions corresponding to the original and deformed tunnel boundaries are as follows: where the circles ζ k � 1 correspond to the circle ζ � 1 and ζ k ⟶ ∞ when ζ ⟶ ∞.
To ensure that the transformed equations (19) and (20) are single valued, all roots of dz k /dζ k � 0 must be located inside the unit circle |ζ k | � 1, thereby giving which should be located inside the unit circle. us, one-toone mapping is obtained. Using equations (17) and (20), z k can also be expressed as By comparing the coefficients of equations (20) and (22) for the tunnel boundary (ρ � 1), one can obtain c kn and d kn . Hence, For n ≥ 2, Similarly, For n ≥ 2, Φ k (z k ) is assumed to have the following form: e displacement components on the tunnel boundary can be obtained using equation (16): It is assumed that the displacement vectors along the tunnel boundary are directed toward the center of the tunnel. e displacement u k at the tunnel boundary can be expressed as e total displacement u k at the tunnel boundary ρ � 1 can also be expressed as Equating the coefficients for similar terms of e inθ and e − inθ in equations (29) and (30) yields e above equations are satisfied by It can be seen that once the mapping function for the original and deformed tunnel boundaries and the parameters of elastic anisotropic soil are determined, the coefficients of equation (27) can be ascertained. e elastic solutions around the tunnel can be obtained using equations (12) and (13).

Validation of the Proposed Solution with
Known Solutions 4.1. Circular Tunnel. For a cylindrical cavity of radius R in infinite anisotropic soil, the boundary condition can be solved by mapping onto a circle of unit radius: e conformal mapping function of z k can be written as When undergoing uniform convergence u ε , the deformed tunnel boundary can be expressed as e conformal mapping function of z k can be written as When subjected to the ovalization mode u δ , the deformed tunnel boundary can be expressed as e conformal mapping function of z k can be written as

Advances in Civil Engineering
e expression for the displacement can be obtained using equations (13) and (32), which is identical to the solution obtained by Zymnis [22].

Nonelliptical Tunnels.
It is useful to determine the coefficients of approximate polynomial mapping functions. is problem has been previously addressed in literature. Heller [27] provided the mapping function for a rectangular opening of unit width and height K, using the Schwarz-Christoffel integral: For nonelliptical holes with rounded corners, we have the following approximate polynomial mapping functions [28]: Manh [13] proposed that, to reduce the error for 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 original and deformed shapes of the hole boundary is obtained, the elastic solution can be determined using the analytical method proposed herein.

Solutions for an Unlined Elliptical Tunnel.
Assuming that the ratio of the semi-major and semi-minor axes remains unchanged during uniform radial convergence of an elliptical tunnel boundary, as shown in Figure 2, we observe that where a 1 , a 2 are the semi-major axes of the elliptical tunnel before and after deformation, respectively; b 1 , b 2 are the semi-minor axes of the elliptical tunnel before and after deformation, respectively; ρ 1 , ρ 2 are the extreme diameter of any point on the elliptical tunnel before and after deformation, respectively; and k is a length ratio.
For an elliptical cavity illustrated in Figure 2 in an infinite medium undergoing uniform convergence, u x and u y , the displacement components at the tunnel wall, can be expressed by e mapping function transforms the elliptical cavity to a unit circle as follows: ereafter, we introduce the following mapping functions, z k (ζ k ) and Z k (ζ k ): 6 Advances in Civil Engineering e coefficients A 1n and A 2n can be obtained from equation (32): When a 1 � b 1 , the results can be reduced to the results obtained by Zymnis [22].

Comparison with Isotropic Soil.
In general, tunnel contours comprise complex curves; thus, to generalize these contours, it is reasonable to simplify complicated tunnels as unlined tunnels with an elliptical outline [29]. To verify the proposed method, we utilized approximately equal stiffness parameters in anisotropic soil; i.e., 335. In addition, we compared the stress and displacement obtained with the proposed theoretical method with the results obtained by the method used in isotropic soil. Figures 3(a)-3(e) compare the distribution of elastic solutions around an elliptical tunnel with a/H � 0.1064; b/a � 0.8. e analyses can be applied under the assumption of incompressible behavior, when the soil is soft clay. e conditions for incompressibility have been given by Gibson [27] as v vh � 0.5, v hh � 1 − 2nv 2 vh � 1 − (n /2). Because the distribution of displacement does not depend on the magnitude of the elliptical tunnel and the distribution of stress does not depend on the magnitude of the modulus, the displacement and stress fields are given by the dimensionless coordinates, x/H and y/H. e results for isotropic soil are shown for x/H < 0, and the results for transversely isotropic soil are shown for x/H > 0 in the following figures. In all the cases, the normal stress and horizontal displacement are symmetric with respect to the x axis, whereas the shear stress and vertical displacement are antisymmetric with respect to the y axis. Excellent agreement is observed between the results for isotropic and transversely isotropic soils, verifying that the anisotropic solutions obtained by the analytical method proposed herein can converge to the isotropic solution with appropriate stiffness parameters.

Comparison of Results for Isotropic Soil and London
Clay. Gasparre [28] noted that London clay exhibits significant anisotropy at low strain levels, i.e., <0.001%.

Virtual Image
Technique. e tunnel is embedded at a depth H from the ground surface. In accordance with the study by Sagaseta, the ground movements associated with an unlined tunnel located at a depth H below the traction-free ground surface can be approximately represented by a Advances in Civil Engineering 9 singularity superposition technique ( Figure 5). e displacement fields caused by a point source/sink (0, − H) in a full plane and an image source (0, +H) of equal and opposite deformation located equidistant above the ground surface are as follows: e resulting normal and shear tractions along the bisecting line y � 0 (simulating a traction-free ground surface) produced by two mirror images are as follows: A distribution of corrective shear tractions along the bisecting line is obtained: e displacement results can be obtained from the following formula: where Φ c 1 (z 1 ) and Φ c 2 (z 2 ) are stress potential functions produced by corrective shear tractions. e final expression for displacement field is determined by combining equations (46) and (49): is expression is still an approximate solution as two types of tunnels have not been considered.

Effect of Anisotropic Stiffness Ratios on the Predicted Displacement.
We examined the effect of the anisotropic stiffness parameters, n (n � E h /E v ) and m (m � G vh /E v ), on the predictions of the ground settlement trough and subsurface horizontal displacement curves for a point offset at a distance x � 2a from the center of the elliptical tunnel. ese results correspond to the solutions for an elliptical tunnel with a/H � 0.1064, b/a � 0.8, and v vh � v hh � 0.5. Figures 6 and 7 show the effect of the anisotropic stiffness ratios, m and n, on the normalized ground displacement and underground horizontal displacement at a reference vertical offset, x � 2a. e results indicate that the stiffness parameters, m and n, have a significant influence on the shape of the displacement distribution. In Figure 7(b), u y0 is the maximum vertical ground displacement calculated for varying values of the anisotropic stiffness parameter m, with n � 1. e point corresponding to y � H is the peak value when m ≤ 0.355, and the solutions generate narrower troughs when m � 0.01. e shape of the displacement trough also changes for m > 0.355, exhibiting double peaks.
In Figure 7, u y0 is the maximum vertical ground displacement calculated for varying values of the anisotropic stiffness parameter n, with m � 0.335. It can be seen that increasing the stiffness parameter n has the opposite effect compared with that for m.
e results indicate narrow troughs when n ≥ 1 (such as London clay), particularly for n > 3. In contrast, for soils with n < 1, double peaks are observed in the curves.

Conclusions
is paper has proposed an analytical method for modeling the displacement due to an unlined tunnel with an arbitrary cross section in an anisotropic plane. For this method, the following features are critical: (1) the original and deformed tunnel boundaries can be written in the form of exact or approximate polynomial mapping functions in the full plane, and the inverse mapping function must be single valued; (2) the solutions describe elastic solutions for soil under the assumption that the displacement vectors along the tunnel boundary are directed toward the center of the Advances in Civil Engineering tunnel. By introducing parameters of elastic anisotropic soil, this method can easily provide elastic solutions for a tunnel with an arbitrary cross section. Several examples are used to illustrate the proposed method, and the case of an elliptical tunnel is discussed in detail to verify the accuracy of the proposed method in full plane. e anisotropic parameters of London clay have negligible influence on the horizontal stress, horizontal displacement, and shear stress; however, these parameters lead to a slightly faster attenuation of the vertical stress and vertical displacement with distance. Surface and ground displacement in half plane can be obtained using a virtual image technique. Consequently, the method developed by Zymnis [22] can be extended from a circular tunnel to a tunnel with an arbitrary cross section.
Our results clearly show that the anisotropic stiffness parameters, m and n, significantly influence the predicted displacement patterns for an unlined tunnel with a certain shape (b/a) and relative embedded depth (a/H).