Explicit Peck formula applied to ground displacement based on an elastic analytical solution for a shallow tunnel

Abstract

Using the complex variable method, an elastic analytical solution of the ground displacement caused by a shallow circular tunneling is derived. Non-symmetric deformation relative to the horizontal center line of the tunnel cross-section is used as a boundary condition. A comparison between the proposed analytical method and the Finite Element Method is carried out to validate the rationality of the obtained analytical solution. Two parameters in the Peck formula, namely the maximum settlement of the ground surface center and the width coefficient of settlement curve, are fitted and determined. We propose a modified Peck formula by considering three input parameters, namely the tunnel depth, tunnel radius, and the tunnel gap. The influence of these three parameters on the modified Peck formula is analyzed. The applicability of the modified Peck formula is further investigated by reference to six engineering projects. The ground surface displacement obtained by the explicit Peck formula is in good agreement with the field data, and the maximum error is only 1.3 cm. The proposed formula can quickly and reasonably predict the ground surface settlement caused by tunnelling.

Graphical abstract

Keywords

complex variable method / elastic analytical solution / maximum settlement formula / modified Peck formula

1 Introduction

The construction of a shallow tunnel generally passes close to existing structures and underground pipelines. The civil engineers responsible for the design and construction should have basic tools to predict the ground deformation and to then evaluate the impact of the ground deformation on existing structures and underground pipelines [1,2]. Four methods can be adopted to predict the ground deformation, namely, numerical methods [3–6], model test methods [7,8], analytical methods [9–11] and empirical methods [12]. The model test methods [13] can intuitively reproduce the ground deformation trends caused by tunnelling. However, it is difficult for the similarity ratio of the geometrical parameters and material mechanical parameters to simultaneously meet the requirements. With respect to complex geological conditions or the complex geometrical shape of a tunnel, the numerical methods [14] provide effective support to predict the ground displacements. However, the numerical methods have apparent drawbacks [15]: 1) the modeling and computing time are substantial; 2) repeated modeling is required in the concept stage of preliminary design; 3) the parameters in the constitutive model are determined according to the borehole exploration at certain positions in the construction field, which cannot reasonably reflect all of the ground conditions. Two alternatives are offered by analytical methods and empirical methods. The coupling effects of the geometrical parameters and geological parameters are quantitatively taken account in the analytical methods [16,17], and these methods can provide guidance for the selection of tunnel structural parameters during the conceptual stage of preliminary tunnel design. An empirical method [12,18,19], namely, the Peck formula expressed by a Gaussian function, has been widely applied by civil engineers and can effectively determine ground settlement trends. The motivation of this work is to obtain an explicit Peck formula by a combination of analytical solutions, then to study the ground deformation trends due to tunneling.

Analytical methods for predicting ground movements have four categories: the Airy stress function elastic theory [20,21], bipolar coordinate method [22], virtual image technique [16], and complex variable method (CVM) [15,23,24]. The Airy stress function is only applicable to mechanical models of deep tunnels. CVM is applicable to both deep and shallow tunnels and it overcomes calculation complexity of the bipolar coordinate method, solving the singularity assumption of the virtual image method. In addition, CVM is characterized by a closed-form solution and high precision [25]. After Verruijt [17,26] considered the stress and displacement boundary of the tunnel perimeter to present the elastic solution from the series convergence of the CVM, the method has been widely used to determine the elastic solution for the ground displacement and stress related to shallow tunnels [27–29]. In the CVM, the biharmonic function that includes the equilibrium equation, geometry equation and elastic stress-strain relationship is expressed by two complex potential functions. Based on reasonable boundary conditions, two complex potential functions expressed in the Laurent series are determined. The stress and the displacement field are presented by determined complex potential functions. Therefore, reasonable boundary conditions are important factors for the elastic solution. Verruijt [26] set the uniform radial displacement as the tunnel displacement boundary condition (DBC) to present the solution of the ground displacement. After that, the “buoyancy effect” was introduced into the analytical solution by adding logarithmic terms into the complex potential functions in the work by Verruijt and Strack [30]. The characteristics of a uniform radial displacement are as follows: the radial displacement magnitudes at any position on the tunnel perimeter are the same, and the direction points toward the tunnel center. However, this idealized deformation pattern cannot reasonably and accurately reflect the main deformation characteristics of the tunnel. A nonuniform deformation pattern is proposed [15,21,25,31–33], including three basic deformation patterns, namely, a uniform radial convergence pattern, ovalization deformation pattern and vertical translation pattern. The nonuniform deformation pattern reflects the non-symmetric deformation relative to the horizontal line. Wang et al. [32] combined a nonuniform deformation pattern and the CVM to successfully predict the ground surface vertical settlement and horizontal displacement in five engineering cases. Fu et al. [33] and Zhang et al. [25] achieved an analytical solution considering the nonuniform deformation mode of a tunnel, and the validity of the proposed methods was verified by several sets of field data. When the geological and construction conditions are basically symmetric relative to the vertical central line of tunnel, the nonuniform deformation pattern can capture the main deformation characteristics of the tunnel cross-section. In our previous work, a unified displacement function represented by the second-order Fourier series was proposed to deal with the nonsymmetric deformation problem, and the work included five basis deformation patterns [27–29,34]. The analytical solutions obtained by the strict mathematical deduction reasonably and quantitatively predict ground displacements. However, the analytical solutions for ground displacement for a shallow tunnel are the implicit expressions including a series of linear equations. The results are only obtained by means of computational software, such as MATLAB. Therefore, the analytical methods cannot be simply and directly used by civil engineers in the field, which limits the application of analytical works in tunnel engineering, to some extent. The Peck formula as an explicit expression is extensively used to predict the ground vertical settlement induced by tunneling. The curve shape of the ground surface settlement can be reasonably described by the Gaussian function, where there are two parameters: maximum settlement of ground surface center s_max and width coefficient of settlement shape i. For certain tunnel projects, some empirical modified Peck formulas have been proposed, incorporating different determination forms for s_max and i [19,35–38]. However, the proposed empirical formulas are based on limited data from the specified tunnel engineering field or model test, and so they cannot be widely applied to other tunnel projects.

The elastic solution of the ground displacement is applied where the nonuniform deformation pattern is used as the tunnel DBC. Taking account of the effects of different tunnel depth, different tunnel radius and different gap parameter, a series of the ground surface vertical settlement curves are presented based on obtained analytical results. The Peck formula is used to fit these ground surface settlement curves. The values of s_max and i considering the depth and radius of the tunnel and the gap parameter are obtained. In this work, the validity of the modified explicit Peck formula is examined by reference to six engineering cases.

2 Analytical method of ground displacement

The boundary conditions are important factors in considering the elastic behavior of a tunnel. A unified displacement function describes the deformation behaviors of a tunnel, as expressed in Eq. (1) [28,29]. Displacements on any position of the tunnel cross-section are known and definite values. Displacement function of the tunnel cross-section satisfies Dirichlet condition, namely, for one cycle: 1) the displacement function is integrable, 2) there is a finite number of maxima or minima; 3) the displacement function is continuous without interruption. Based on the abovementioned condition, the displacement function of the tunnel cross-section can be expressed in a Fourier series.

(1)u_r\left({\theta }^{\prime }\right)=-u_0+a_1^{\prime }{\sin \theta }^{\prime }+b_1^{\prime }{\cos \theta }^{\prime }+a_2^{\prime }\sin 2{\theta }^{\prime }+b_2^{\prime }\cos 2{\theta }^{\prime },

where u_r\left({\theta }^{\prime }\right) is unified displacement function, u₀ is constant term, a_1^{\prime }, b_1^{\prime }, a_2^{\prime }, and b_2^{\prime } are coefficients of Fourier trigonometric series.

Unified displacement function (1) can simultaneously describe non-symmetric deformation relative to both horizontal centerline (x^{\prime } axis in Fig.1) and vertical centerline (y^{\prime } axis in Fig.1). This work focuses on the research of engineering problems under non bias loads. Under this condition, scholars have summarized that the deformation of the tunnel cross-section is symmetric relative to the vertical centerline and non-symmetric relative to the horizontal centerline [11,21]. The non-symmetric deformation items relative to the vertical centerline, namely b_1^{\prime } and a_2^{\prime }, are equal to 0. Equation (1) is reduced to Eq. (2).

(2)u_r\left({\theta }^{\prime }\right)=-u_0+a_1^{\prime }{\sin \theta }^{\prime }+b_2^{\prime }\cos 2{\theta }^{\prime }.

There are three deformation parameters in Eq. (2), i.e., u_0, a_1^{\prime }, and b_2^{\prime }. u_0 describes the ground loss. a_1^{\prime }{\sin \theta }^{\prime } and b_2^{\prime }\cos 2{\theta }^{\prime } reflect the vertical translation and ovalization deformation of a tunnel as shown in Fig.1, respectively. Some scholars take Eq. (2) as the DBC in the analytical studies [25,32,33,39,40]. In these works, these three deformation parameters are unknown.

Park proposed a deformation pattern of the tunnel in Eq. (3) [21]. When Eq. (3) is used as the DBC, the Airy stress function method is used to present the elastic solution of the ground displacement for the infinite domain analysis model. In particular, the infinite domain model ignores the influence of the ground surface; therefore, the analytical results obtained by Park [21] cannot reasonably reveal the settlement laws of the ground surface.

(3)u_r\left({\theta }^{\prime }\right)=-\frac{u_0^{\prime }}{4}\left(5+3{\sin \theta }^{\prime }-3{{\cos }^2\theta }^{\prime }\right).

Equation (3) can be also written as,

(4)u_r\left({\theta }^{\prime }\right)=-\frac{7u_0^{\prime }}{8}-\frac{3u_0^{\prime }}{4}{\sin \theta }^{\prime }+\frac{3u_0^{\prime }}{8}\cos 2{\theta }^{\prime }.

Equation (4) is a special case of Eqs. (1) and (2). In particular, there is only an unknown parameter in Eq. (4), namely u_0^{\prime }. Lee et al. [41] defined the gap parameter g as a measure of the magnitude of the equivalent two-dimensional void formed around the tunnel caused by over-excavation of tunnel cross-section creating a physical gap that is related to the shield machine, the lining and surrounding ground. For the displacement boundary function (4), the relation between u_0^{\prime } and g [32] is,

(5)u_0^{\prime }=\frac{8}{7}g.

Equation (4) as the DBC can well describe deformation characteristics of the tunnel [11,15,21,31].

The analytical mechanical model of a shallow tunnel is represented in Fig.2. r₀ and h are the tunnel radius and the center depth, respectively. z is any point in the z plane and is denoted as x + yi. z plane is the physical plane and ζ plane is the complex potential plane. R region in the z plane is mapped to γ₁ region in the ζ plane. u_x and u_y are horizontal and vertical displacement, respectively. σ_xx and σ_yy are horizontal and vertical normal stress, respectively. The elastic analytical solution of ground displacement is derived based on the following assumptions: the ground is considered to be a homogeneous isotropic elastic material, and the shape of the tunnel cross-section is circular.

Zero stress and Eq. (4) are respectively set as the boundary conditions of the ground surface and the tunnel, in Eqs. (6) and (7),

(6)\mathit{\text{z}}=\overline{\mathit{\text{z}}}:\phi \left(\mathit{\text{z}}\right)+\mathit{\text{z}}\overline{{\phi }^{\prime }\left(\mathit{\text{z}}\right)}+\overline{\psi \left(\mathit{\text{z}}\right)}=0,

(7)\left|\mathit{\text{z}}+\mathrm{i}h\right|=r_0:2G(u_x+\mathrm{i}{u}_y)=\kappa \phi \left(\mathit{\text{z}}\right)-\mathit{\text{z}}\overline{{\phi }^{\prime }\left(\mathit{\text{z}}\right)}-\overline{\psi \left(\mathit{\text{z}}\right)},

where \phi \left(\mathit{\text{z}}\right) and \psi \left(\mathit{\text{z}}\right) are complex potential functions [42]. \nu and G are Poisson’s ratio and shear modulus of the ground, respectively. \kappa =3-4\nu.

By the conformal mapping function, position coordinates in the physical plane can be expressed as,

(8)\mathit{\text{z}}=w\left(\zeta \right)=-\mathrm{i}\eta \frac{1+\zeta }{1-\zeta },

where \eta =h{\displaystyle \frac{1-{\alpha }^2}{1+{\alpha }^2}}, {\displaystyle \frac{2\alpha }{1+{\alpha }^2}}={\displaystyle \frac{r_0}{h}}. The circles with radii 1 and α in the complex plane represent the ground surface and the circular tunnel in the physical plane, respectively.

Complex potential functions can be expressed as follows by the Laurent series.

(9)\phi \left(\zeta \right)=a_0+\sum _{k=1}^{\mathrm{\infty }}{a}_k{\zeta }^k+\sum _{k=1}^{\mathrm{\infty }}{b}_k{\zeta }^{-k},

(10)\psi \left(\zeta \right)=c_0+\sum _{k=1}^{\mathrm{\infty }}{c}_k{\zeta }^k+\sum _{k=1}^{\mathrm{\infty }}{d}_k{\zeta }^{-k},

Equations (8)–(10) are substituted into Eq. (6), and the following relation can be presented.

(11)c_0=-\overline{a_0}-\frac{1}{2}{a}_1-\frac{1}{2}{b}_1,

(12)c_k=-\overline{b_k}+\frac{1}{2}(k-1)a_{k-1}-\frac{1}{2}(k+1){a_{k+}}_1,k=1,2,3,...,

(13)d_k=-\overline{a_k}+\frac{1}{2}(k-1)b_{k-1}-\frac{1}{2}(k+1){b_{k+}}_1,k=1,2,3,....

Considering the Eqs. (7)–(13), the following relation is determined,

(14)\begin{array}{rl}& (1+\kappa {\alpha }^{2k+2})a_{k+1}+(1-{\alpha }^2)(k+1)\overline{b_{k+1}}\\ =& \left\{{\alpha }^2(1+\kappa {\alpha }^{2k})a_k+(1-{\alpha }^2)k\overline{b_k}+A_{k+1}{\alpha }^{k+1}\right\},\\ & k=1,2,3,...,\end{array}

(15)\begin{array}{rl}& (1-{\alpha }^2){\alpha }^{2k}(k+1)a_{k+1}-(\kappa +{\alpha }^{2k+2})\overline{b_{k+1}}\\ =& \left\{(1-{\alpha }^2){\alpha }^{2k}k{a}_k-(\kappa +{\alpha }^{2k})\overline{b_k}+\overline{A_{-k}}{\alpha }^k\right\},\\ & k=1,2,3,...,\end{array}

(16)(1-{\alpha }^2)\overline{a_1}-(\kappa +{\alpha }^2)b_1=A_0-(\kappa +1)a_0,

(17)(1+\kappa {\alpha }^2)a_1+(1-{\alpha }^2)\overline{b_1}=A_1\alpha +(\kappa +1){\alpha }^2{a}_0,

where A₀, A₁, A_k+1, and A_−k are the series expansion expressions of Eq. (4). The coefficients of the complex potential function are expressed by iterative equations in Eqs. (14)–(17). As the number k of iterations increases, a_k, b_k, c_k, and d_k gradually approach 0. The termination condition of iterative calculation in this work is that a_k, b_k, c_k, and d_k are less than 10^{-8}. Once a_k, b_k, c_k, and d_k are determined, two complex potential functions \phi \left(\mathit{\text{z}}\right) and \psi \left(\mathit{\text{z}}\right) are obtained. The horizontal and vertical displacement at different positions of the ground can be obtained by Eq. (18).

(18)u_x+\mathrm{i}{u}_y=\frac{\kappa \phi \left(\mathit{\text{z}}\right)-\mathit{\text{z}}\overline{{\phi }^{\prime }\left(\mathit{\text{z}}\right)}-\overline{\psi \left(\mathit{\text{z}}\right)}}{2G}.

a₀ in Eqs. (16) and (17) is the rigid body displacement term and can be determined by the series convergence [33,34]. If the weight of the tunnel lining is less than that of the excavated soil, an upthrust force is generated acting on the tunnel cross-section. The influence of the upthrust on the ground response caused by tunneling is a “buoyancy effect” . In this work, the weight of the tunnel lining is considered equal to that of the excavated soil, and an upthrust is equal to 0 [30,43]. Under this condition, the ground displacement is independent of the shear modulus G [44].

To verify the validty of the proposed method, Finite Element Method (FEM) is adopted to compare the analytical solution with the numerical results. Finite element software ABAQUS 6.14 is applied to establish the numerical model. The model size of the FEM is 80 m × 80 m, as shown in Fig.3(a). The quadrilateral reducted integral method is used to divide FEM meshes. The plane strain condition is adopted, which is consistent with the analytical model. Poisson’s ratio and Young’s modulus are 0.3 and 30 MPa, respectively. The center depth and radius of the tunnel are 10 and 3 m, respectively. The top, bottom, and both sides of the FEM model in Fig.3(a) are free boundaries. The employed geometric relationship can eliminate the influence of boundary effects on the calculation results. 40 meshes are used to divide the tunnel profile, and each node corresponds to 0°, 9°, 18°, …, 360°. When the displacement boundary function Eq. (4) is given, the displacement of each node can be then determined. The displacement value of each node is edited into the “Edit keywords” in ABAQUS as the DBC. For a given DBC, the ground surface in the range of 40 m is selected to compare the proposed method with the FEM, as the red line shown in Fig.3(a). When u_0^{\prime } is equal to 0.05 and 0.10 m, respectively, the ground surface settlement found by two methods is as shown in Fig.3(b). The results by the proposed method and FEM are quantitatively consistent, which presents the rationality of the analytical solution.

When r₀ = 3 m, h = 10 m, ν = 0.3; the influence of u_0^{\prime } on the surface vertical and horizontal displacement is shown in Fig.4. As shown in Fig.4, for any position of the ground surface, the vertical settlement and the absolute value of horizontal displacement are always larger for a larger value of u_0^{\prime }. Horizontal displacements of surface center are equal to 0 in Fig.4(b). The influence of u_0^{\prime } on the ground vertical settlement in different depths is shown in Fig.5. h' = h – r₀ = 7 m. As the depth increases, the maximum vertical settlement of the center point increases and the width of the settlement trough curve decreases. For the same depth, the ground vertical settlement is larger for a larger value of u_0^{\prime } in Fig.5.

3 Modified Peck curve of the ground surface settlement

The distribution laws of ground surface displacement caused by influences of tunnel depth, tunnel radius, and ground loss are studied. The Peck formula is used to fit these large amounts of analytical calculation data. Empirical formulas for s_max and Peck curves are presented below.

3.1 Empirical formula for the maximum settlement of the surface center point

There are two parameters in the Peck formula in Eq. (19), namely, s_max and i. These two unknown parameters depend on the tunnel geometrical parameters, the material mechanical parameters and the deformation parameters of the tunnel perimeter.

(19)s\left(x\right)=s_{max}{\mathrm{e}}^{\left(-\frac{x^2}{2i^2}\right)},

where x is the distance between the observation point and the center point of the ground surface. In actual tunnel engineering, the Poisson’s ratio of the ground is difficult to determine, and the Poisson’s ratio is assumed to be 0.3 in this work. Therefore, based on Eqs. (6)–(18), the values of s_max and i are related to r₀, h, and u_0^{\prime }.

The curves of the ground surface vertical settlement for different values of h are shown in Fig.6, for relative depths (r₀/h) of 1/4 and 1/5. As we can see, the curve shape of the vertical settlement presented by the proposed analytical solution fits the Gaussian curve shape. s_max is always the same when the relative depth r₀/h is the same. The value of s_max is larger corresponding to a larger r₀/h as shown in Fig.6(a) and Fig.6(b). s_max is related to r₀/h and u_0^{\prime } as expressed in Eq. (20),

(20)s_{max}=f\left(u_0^{\prime },r_0/h\right).

The following conditions are designed to obtain the value of s_max, namely, that the values of u_0^{\prime } are equal to 0 , 0.05 , 0.10 , 0.15 , 0.20 , 0.25 , 0.30 , 0.35 , 0.40 , 0.45 , and 0.50 m; and the corresponding values of r₀/h are equal to 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8. As the value of u_0^{\prime } increases, the value of s_max increases, as shown in Fig.7. When the values of u_0^{\prime } are the same, s_max is larger for a larger value of r₀/h. A linear relation can be seen between s_max and r₀/h, and the slope of each line is related to u_0^{\prime }. As shown in Fig.7, the value of the line slope is small, corresponding to a large value of u_0^{\prime }. Equation (21) is established to describe the relationships between s_max, u_0^{\prime }, and r₀/h. The range of R² of the ten lines shown in Fig.7 is 0.99270 to 0.99276 and mean value of R² is 0.99272.

(21)s_{max}=\frac{3u_0^{\prime }{r}_0}{h}.

Equation (21) is obtained on the basis of the 81 conditions shown in Fig.7. An additional 18 conditions are selected to verify the validity of Eq. (21), which are different from the abovementioned 81 conditions. r₀/h is equal to 0.15, 0.25, 0.35, 0.45, 0.55, and 0.65. For each r₀/h, there are 3 values of u_0^{\prime }: 0.075, 0.275, and 0.475 m. The results comparing the analytical solution and Eq. (21) are presented in Fig.8. The results from Eq. (21) are in close agreement with the results of the proposed analytical solution in Section 2.

3.2 Modified Peck curve of the ground surface settlement

The ground surface settlement Peck curve is expressed in Eq. (22) by combining Eq. (19) with Eq. (21). Equation (22) is used to describe the ground surface vertical settlement curves derived by the analytical method and then to determine relations among i, r₀, h, and u_0^{\prime }. Finally, the modified Peck curve is obtained, and a series of parameter analyses are investigated.

(22)s\left(x\right)=-\frac{3u_0^{\prime }{r}_0}{h}{\mathrm{e}}^{\left(-\frac{x^2}{2i^2}\right)}.

Four different tunnel geometry conditions are designed to study the influence of u_0^{\prime } on the value of i: r₀ = 2 m, h = 6 m; r₀ = 2 m, h = 10 m; r₀ = 4 m, h = 6 m; and r₀ = 4 m, h = 10 m. For each tunnel geometry condition, ten different values of u_0^{\prime } are considered from 0.05 to 0.5 m. There are thus 40 conditions. Equation (22) is adopted to fit and obtain the value of i for each condition in Fig.9. When the values of r₀ and h are the same, respectively, the value of u_0^{\prime } does not influence the value of i. For the same values of h, r₀ has less influence on i. For the same values of r₀, h has a significant influence on the value of i.

Eight different tunnel radii are considered, namely, r₀ = 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, and 5.5 m. For r₀ = 2.0, 2.5, 3.0, and 3.5 m, h ranges from 4 to 12 m. For r₀ = 4.0 m, h ranges from 5 to 12 m. For r₀ = 4.5, 5.0, and 5.5 m, h ranges from 6 to 12 m. The increment in h is 1.0 m. There are 65 conditions to study the influence of r₀ and h on i. As shown in Fig.10, for the same r₀, the value of i linearly increases with the increase in h. The relation curve between i and h is fitted for each r₀ by Eq. (23). The fitted results are shown in Fig.10 and Tab.1. The values of R² for the eight curves are greater than 0.987, indicating that the curves are reasonably fitted to the linear relation:

(23)i=k_i\cdot h.

As r₀ increases, the value of k_i decreases. The relation between r₀ and k_i is expressed as shown in Eq. (24)

(24)k_i=-0.0127r_0+0.8240.

Combining Eqs. (22) and (24), the modified Peck curve of the ground surface settlement is obtained as follows.

(25)s\left(x\right)=-\frac{3u_0^{\prime }{r}_0}{h}{\mathrm{e}}^{\left[-\frac{x^2}{2{\left(-0.0127r_0+0.8240\right)}^2{h}^2}\right]}.

Explicit Peck Eq. (25) is obtained by fitting the data in the elastic analytical solution of shallow circular tunnel. When values of r₀, h, and u_0^{\prime } are known for a specific tunnel, the ground surface settlement can be presented by Eq. (25). The deformation parameter u_0^{\prime } of tunnel cross-section comprehensively reflects the impact of construction parameters.

Three conditions are considered to further verify the validty of the modified Peck curve by comparing Eq. (25) with the analytical solution. When u_0^{\prime } = 0.15 m and h = 10 m, the values of r₀ are considered to be equal to 1, 3, and 5 m and the settlement curves from the two methods are shown in Fig.11. The settlement curves resulting from Eq. (25) and from the analytical solution are nearly identical for the same values of r₀, which indicates the validity of Eq. (25).

There are three parameters in the modified Peck curve in Eq. (25), namely, r₀, h, and u_0^{\prime }. The influences of the three parameters are discussed as follows. When r₀ = 3 m and h = 8 m, the influence of u_0^{\prime } on Eq. (25) is as shown in Fig.12(a). The settlement at any point of the ground surface gradually becomes larger as the value of u_0^{\prime } increases. The settlement approaches 0 in almost the same position for different values of u_0^{\prime }. When u_0^{\prime } = 0.1 m and h = 8 m, the influence of r₀ on Eq. (25) is shown in Fig.12(b). The value of r₀ simultaneously influences the maximum value and width coefficient of the settlement curve. The settlement for any point of the ground surface increases with the increase of r₀. When u_0^{\prime } = 0.1 m and r₀ = 3 m, the influence of h on Eq. (25) is shown in Fig.12(c). The value of h has a significant influence on the shape of the settlement curve. As the value of h decreases, the settlement of the surface center point increases. However, when the distance from the center of the tunnel is small, the settlement corresponding to the large value of h is small. As the distance increases, the value of s(x) becomes larger for a larger value of h.

4 Case studies

To assess the potential of the modified Peck formula relating to the ground surface settlement and to compare the ground surface settlement obtained by Eq. (25) with the field data, six sets of tunnel engineering problems are chosen,. The engineering applications include the Bangkok Tunnel [32], Docklands Light Railway Lewisham Extension (MS-5) [33], Frankfurt Subway Tunnel [33], Green Park Tunnel [32], Heathrow Express Trial Tunnel [32], and Thunder Bay Tunnel [32]. The new Austrian tunnelling method is used in the Heathrow Express Trial Tunnel, and the shield method is applied in the other five tunnels.

The three parameters in this work, namely, r₀, h, and u_0^{\prime }, are known in the six tunnel engineering projects. In addition, the ground conditions are provided in Tab.2. The predicted curve of the ground surface settlement can be obtained for each tunnel when the values of r₀, h, and u_0^{\prime } are substituted into Eq. (25). The comparison results between the predicted curve and the field data are presented in Fig.13. Although the predicted values of the surface vertical settlement obtained by Eq. (25) can slightly overestimate or underestimate the value obtained from the field data, the shape of the predicted curve is basically in agreement with the field settlement trends. The ground surface settlement shows that the position corresponding to the maximum ground surface settlement is at the ‘center point’ (that is, directly above the central axis of the tunnel) of the ground surface. The settlement value decreases rapidly as the distance from the center point increases, and the settlement decreases slowly when the distance is larger than the width coefficient of the settlement curve. As shown in Fig.13(f), there are somewhat difference between the field data and the proposed method for the Thunder Bay Tunnel case. The shield method is applied in the Thunder Bay Tunnel, and “during the advance a clay grout is injected into the tailpiece void” [45]. Although significant ground losses occur at the tunnel cross-section, clay grout can effectively prevent the expansion of ground loss to the ground surface. Therefore, only considering the ground loss u_0^{\prime } and substituting it into the explicit Peck formula, the calculated ground surface displacement is greater than the field data.

5 Conclusions

Based on the Laurent series and conformal mapping function of the CVM, the elastic analytical solution for a shallow tunnel is achieved where non-symmetric deformation relative to the horizontal center line is the DBC of the tunnel. The value of the surface vertical settlement by the proposed analytical method is quantitatively consistent with the results obtained by FEM, where the tunnel geometry, geological parameters and boundary conditions are the same for the two methods.

The displacement results of the shallow tunnel based on the CVM are the implicit expressions. The results can only be determined by means of the mathematical calculation software, which is hard to use by researchers or civil engineers who are not working on the analytical solutions. To obtain an explicit expression of the ground surface vertical settlement curve, the Peck formula is used to fit the vertical settlement values by the analytical method. The parameters of the analytical solution are denoted as s_max and i of the Peck formula. The values of r₀/h and u_0^{\prime } has an influence on the value of s_max. s_max becomes large linearly with the increase of r₀/h or u_0^{\prime }. The expression relating r₀/h, u_0^{\prime }, and s_max together is established. The value of u_0^{\prime } doesn't affect the value of i. The value of i is significantly affected by h. r₀ has less effect on i. The value of i is smaller for a larger value of r₀. Combining the proposed expressions of s_max and i, the modified Peck formula is obtained. The influences of r₀, h, u_0^{\prime } on the modified Peck formula are discussed. The proposed modified Peck formula is on the whole close to the field data of six engineering tunnel projects.

u_0^{\prime } value in Eq. (25) is unknown before a tunnel excavation. Machine learning method can identify a nonlinear relationship of high-dimensional parameters based on existing big data, providing an applicable method to predict many problems in geotechnical engineering. In future work, the authors will use machine learning methods to establish a model for predicting the u_0^{\prime } value based on existing large amounts of tunnel engineering data. The proposed method can predict the value of the ground surface vertical settlement quickly and accurately, which can provide theoretical guidance for the selection of tunnel support structure parameters and the evaluation of ground settlement in the concept phase of tunnel design.

References

[1] Ng C, Boonyarak T, Mašín D. Effects of pillar depth and shielding on the interaction of crossing multitunnels. Journal of Geotechnical and Geoenvironmental Engineering, 2015, 141(6): 04015021

[2] Franza A, Marshall A M, Haji T, Abdelatif A O, Carbonari S, Morici M. A simplified elastic analysis of tunnel-piled structure interaction. Tunnelling and Underground Space Technology, 2017, 61(1): 104–121

[3] Zhao K, Janutolo M, Barla G. A completely 3D model for the simulation of mechanized tunnel excavation. Rock Mechanics and Rock Engineering, 2012, 45(4): 475–497

[4] Zhao K, Bonini M, Debernardi D, Janutolo M, Barla G, Chen G X. Computational modelling of the mechanised excavation of deep tunnels in weak rock. Computers and Geotechnics, 2015, 66: 158–171

[5] Zheng G, Zhang T Y, Diao Y. Mechanism and countermeasures of preceding tunnel distortion induced by succeeding EPBS tunneling in close proximity. Computers and Geotechnics, 2015, 66: 53–65

[6] Zhang M J, Li S H, Li P F. Numerical analysis of ground displacement and segmental stress and influence of yaw excavation loadings for a curved shield tunnel. Computers and Geotechnics, 2020, 118: 103325

[7] Zhu W S, Li Y, Li S C, Wang S G, Zhang Q B. Quasi-three-dimensional physical model tests on a cavern complex under high in-situ stresses. International Journal of Rock Mechanics and Mining Sciences, 2011, 48(2): 199–209

[8] Li P F, Zhao Y, Zhou X J. Displacement characteristics of high-speed railway tunnel construction in loess ground by using multi-step excavation method. Tunnelling and Underground Space Technology, 2016, 51: 41–55

[9] Yu H T, Cai C, Guan X F, Yuan Y. Analytical solution for long lined tunnels subjected to travelling loads. Tunnelling and Underground Space Technology, 2016, 58: 209–215

[10] Yu H T, Cai C, Bobet A, Zhao X, Yuan Y. Analytical solution for longitudinal bending stiffness of shield tunnels. Tunnelling and Underground Space Technology, 2019, 83: 27–34

[11] Zhang Z G, Huang M S, Zhang C P, Jiang K M, Wang Z W, Xi X G. Complex variable solution for twin tunneling-induced ground movements considering nonuniform convergence pattern. International Journal of Geomechanics, 2020, 20(6): 04020060

[12] Lu D C, Lin Q T, Tian Y, Du X L, Gong Q M. Formula for predicting ground settlement induced by tunnelling based on Gaussian function. Tunnelling and Underground Space Technology, 2020, 103: 103443

[13] Zheng G, Dai X, Diao Y, Zeng C F. Experimental and simplified model study of the development of ground settlement under hazards induced by loss of groundwater and sand. Natural Hazards, 2016, 82(3): 1869–1893

[14] Hasanpour R, Chakeri H, Ozcelik Y, Denek H. Evaluation of surface settlements in the Istanbul metro in terms of analytical, numerical and direct measurements. Bulletin of Engineering Geology and the Environment, 2012, 71(3): 499–510

[15] Zhang Z G, Pan Y T, Zhang M X, Lv X L, Jiang K M, Li S N. Complex variable analytical prediction for ground deformation and lining responses due to shield tunneling considering groundwater level variation in clays. Computers and Geotechnics, 2020, 120: 103443

[16] Sagaseta C. Analysis of undrained soil deformation due to ground loss. Geotechnique, 1987, 37(3): 301–320

[17] Verruijt A. A complex variable solution for a deforming circular tunnel in an elastic half-plane. International Journal for Numerical and Analytical Methods in Geomechanics, 1997, 21(2): 77–89

[18] PeckR B. Deep excavations and tunneling in soft ground. In: Proceedings of 7th International Conference SMFE. Mexico: A.A. Balkema, 1969, 225–290

[19] Mair R J, Taylor R N, Bracegirdle A. Subsurface settlement profiles above tunnels in clays. Geotechnique, 1993, 43(2): 315–320

[20] Chou W I, Bobet A. Predictions of ground deformations in shallow tunnels in clay. Tunnelling and Underground Space Technology, 2002, 17(1): 3–19

[21] Park K H. Elastic solution for tunneling-induced ground movements in clays. International Journal of Geomechanics, 2004, 4(4): 310–318

[22] Yang X L, Wang J M. Ground movement prediction for tunnels using simplified procedure. Tunnelling and Underground Space Technology, 2011, 26(3): 462–471

[23] Fu J Y, Yang J S, Yan L, Abbas S M. An analytical solution for deforming twin-parallel tunnels in an elastic half plane. International Journal for Numerical and Analytical Methods in Geomechanics, 2015, 39(5): 524–538

[24] Fang Q, Song H R, Zhang D L. Complex variable analysis for stress distribution of an underwater tunnel in an elastic half plane. International Journal for Numerical and Analytical Methods in Geomechanics, 2015, 39(16): 1821–1835

[25] Zhang Z G, Huang M S, Xi X, Yuan X. Complex variable solutions for soil and liner deformation due to tunneling in clays. International Journal of Geomechanics, 2018, 18(7): 04018074

[26] Verruijt A. Deformations of an elastic half plane with a circular cavity. International Journal of Solids and Structures, 1998, 35(21): 2795–2804

[27] Lu D C, Kong F C, Du X L, Shen C P, Gong Q M, Li P F. A unified displacement function to analytically predict ground deformation of shallow tunnel. Tunnelling and Underground Space Technology, 2019, 88: 129–143

[28] Kong F C, Lu D C, Du X L, Shen C P. Displacement analytical prediction of shallow tunnel based on unified displacement function under slope boundary. International Journal for Numerical and Analytical Methods in Geomechanics, 2019, 43(1): 183–211

[29] Kong F C, Lu D C, Du X L, Shen C P. Elastic analytical solution of shallow tunnel owing to twin tunnelling based on a unified displacement function. Applied Mathematical Modelling, 2019, 68: 422–442

[30] Verruijt A, Strack O E. Buoyancy of tunnels in soft soils. Geotechnique, 2008, 58(6): 513–515

[31] Park K H. Analytical solution for tunnelling-induced ground movement in clays. Tunnelling and Underground Space Technology, 2005, 20(3): 249–261

[32] Wang L Z, Li L L, Lu X J. Complex variable solutions for tunneling-induced ground movement. International Journal of Geomechanics, 2009, 9(2): 63–72

[33] Fu J Y, Yang J S, Klapperich H, Wang S Y. Analytical prediction of ground movements due to a nonuniform deforming tunnel. International Journal of Geomechanics, 2016, 16(4): 04015089

[34] Lu D C, Kong F C, Du X L, Shen C P, Su C C, Wang J. Fractional viscoelastic analytical solution for the ground displacement of a shallow tunnel based on a time-dependent unified displacement function. Computers and Geotechnics, 2020, 117: 103284

[35] Marshall A M, Farrell R, Klar A, Mair R J. Tunnels in sands: The effect of size, depth and volume loss on greenfield displacements. Geotechnique, 2012, 62(5): 385–399

[36] Loganathan N, Poulos H G. Analytical prediction for tunneling induced ground movement in clays. Journal of Geotechnical and Geoenvironmental Engineering, 1998, 124(9): 846–856

[37] Lee Y J. Investigation of subsurface deformations associated with model tunnels in a granular mass. Tunnelling and Underground Space Technology, 2009, 24(6): 654–664

[38] Lin Q T, Meng X, Lu D C, Miao J B, Zhao Z H, Du X L. A unified empirical method for predicting both vertical and horizontal ground displacements induced by tunnel excavation. Canadian Geotechnical Journal, 2024, 61(7): 1468–1495

[39] Gonzalez C, Sagaseta C. Patterns of soil deformations around tunnels. Application to the extension of Madrid Metro. Computers and Geotechnics, 2001, 28(6): 445–468

[40] Pinto F, Whittle A J. Discussion of “Elastic solution for tunneling-induced ground movements in clays” by KH Park. International Journal of Geomechanics, 2006, 6(1): 72–73

[41] Lee K M, Rowe R K, Lo K Y. Subsidence owing to tunnelling. I. Estimating the gap parameter. Canadian Geotechnical Journal, 1992, 29(6): 929–940

[42] MuskhelishviliN I. Some Basic Problems of The Mathematical Theory of Elasticity. Groningen: Noordhoff, 1953

[43] Kong F C, Lu D C, Ma Y D, Tian T, Yu H T, Du X L. Novel hybrid method to predict the ground-displacement field caused by shallow tunnel excavation. Science China Technological Sciences, 2023, 66(1): 101–114

[44] Kong F C, Lu D C, Ma C, Shen C P, Yang X D, Du X L. Fractional viscoelastic solution of stratum displacement of a shallow tunnel under the surface slope condition. Underground Space, 2023, 10: 233–247

[45] Palmer J H L, Belshaw D J. Deformations and pore pressures in the vicinity of a precast, segmented, concrete-lined tunnel in clay. Canadian Geotechnical Journal, 1980, 17(2): 174–184

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