Simplified method for evaluating shield tunnel deformation due to adjacent excavation

Tunnelling and Underground Space Technology 71 (2018) 94–105

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Simplified method for evaluating shield tunnel deformation due to adjacent excavation

MARK

⁎

Rongzhu Lianga,b, , Wenbing Wua, Feng Yuc, Guosheng Jianga, Junwei Liub a b c

Engineering Faculty, China University of Geosciences (Wuhan), Wuhan, Hubei 430074, China Research Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou 310058, China School of Civil Engineering and Architecture, Zhejiang Sci-Tech University, Hangzhou 310018, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Adjacent excavation Existing shield tunnel Tunnel response Simplified analytical method

Due to the repaid development of underground space in big cities, increasing excavation pits is being constructed or planned in a close proximity to existing metro tunnels in dense urban areas. Adjacent excavation inevitably changes ground stress state and leads to soil movements around nearby tunnels, which may cause a series of adverse impacts on the underlying existing tunnels. Thus, to evaluate the responses of existing shield tunnels associated with adjacent excavation tunnel is crucial and essential for geotechnical engineers. Current semianalytical methods generally utilize the Winkler foundation with the Vesic′s subgrade modulus to consider tunnel-excavation interactions. However, the Winkler model cannot further consider the interaction between adjacent springs and the Vesic′s subgrade modulus expression is incapable of considering the effect of tunnel embedment depth on the tunnel-ground relative stiffness. In this paper, a simplified analytical method is thus proposed to predict the shield tunnel behaviors associated with adjacent excavation by introducing the Pasternak foundation model with a modified subgrade modulus. The shield tunnel is treated as a continuous Euler-Bernoulli beam resting on the Pasternak foundation model and a modified subgrade modulus expression is presented to consider the effect tunnel embedment depth on subgrade modulus. Two-stage analysis method is applied to analyze the tunnel responses. First, the excavation induced vertical unloading stress acting on the underlying tunnel is calculated via widely-used Mindlin′s solution, ignoring the presence of the existing shield tunnel. Second, the responses of the shield tunnel due to the imposing vertical unloading stress are analyzed using the finite difference method. The feasibility of the proposed method is verified by comparison with the results from a three-dimensional finite element analysis and two published filed measurements. The predicted results are also compared with the results from the Winkler-based method. Finally, parametric studies are performed to investigate the effects of different factors on the responses of existing shield tunnel, including the ground elastic modulus, excavation depth and excavation geometry.

1. Introduction In congested urban cities, metro system plays an extremely important role in city traffic systems and everyday thousands of people travel by metro trains. The safety and serviceability of existing metro tunnels are always under serious concern. Due to the rapid development of underground space in big cities, it is an increasing commercial demand for construction of car parks or underground supermarkets in a close proximity to metro tunnel lines. The adjacent geotechnical engineering activities, such as deep excavation, may cause adverse impacts on or even potential damage to the nearby shield exiting tunnels. If the induced tunnel deformation and internal forces exceed the capacity of tunnel structures, segment cracking, leakage and even

⁎

longitudinal distortion of railway track may likely occur subsequently, which may seriously threaten the smoothness and safety of running trains. Chang et al.(2001) extensively reported a shield tunnel damage case history due to an adjacent deep excavation. Cracks in segmental linings and distortion of connected blots were observed in this case history. Therefore, it is one of major challenges for city designers and geotechnical engineers to evaluate shield tunnel responses associated with an adjacent excavation. Lots of studies have been published to investigate the effects of an adjacent excavation on existing shield tunnels by means of various methods, including field monitoring (Burford, 1988; Chang et al., 2001; Simpson and Vardanega, 2014), centrifuge modelling test (Zheng et al., 2010; Ng et al., 2013; Huang et al., 2014), numerical analysis (Lo and Ramsay, 1991; Dolezalova,

Corresponding author at: Engineering Faculty, China University of Geosciences (Wuhan), Wuhan, Hubei 430074, China. E-mail addresses: [email protected] (R. Liang), [email protected] (W. Wu), [email protected] (F. Yu), [email protected] (G. Jiang), [email protected] (J. Liu).

http://dx.doi.org/10.1016/j.tust.2017.08.010 Received 24 August 2015; Received in revised form 29 November 2016; Accepted 7 August 2017 0886-7798/ © 2017 Published by Elsevier Ltd.

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L

2001; Sharma et al., 2001. Hu et al., 2003; Zheng and Wei, 2008; Liu et al., 2010; Devriendt et al., 2010; Huang et al., 2013; Shi et al., 2015) and semi-analytical methods (Zhang et al., 2013a; Zhang et al., 2013b, 2015). Compared to the high cost filed measurement and centrifuge modelling test and complex finite element model building, analytical method serves as a rapid and low cost approach to estimate the shield tunnel responses to an adjacent excavation at the design and preliminary planning stage. Yet few analytical methods have been proposed (Zhang et al., 2013a; Zhang et al., 2013b, 2015) for estimation of shield tunnel responses due to excavation. Among these previous analytical methods, a existing shield tunnel is commonly treated as a continuous beam resting on a Winkler type foundation and the subgrade modulus is always estimated using the Vesic′s expression (Vesic, 1961). The Winkler model presented by Winkler (1867) is based on the hypothesis that the soil is made up by continuously distributed, nonconnected discrete springs and the pressure at any point on the surface is proportional to the ground deflection. It is expressed as

p = kw (x )

B

h

H

Excavation

Existing shield tunnel

(1)

Fig. 1. Interaction between the excavation and existing shield tunnel.

where p is the pressure at top of spring; k is the coefficient of subgrade modulus; and w(x) is the deflection of beam. However, this model has some shortcomings of the inherent discontinuity of adjacent springs, which cannot perfectly represent the mechanical behavior of foundation material and gives inaccurate prediction of bending moments on beams (Tanahashi, 2004). Besides, the Vesic′s expression is deduced by allowing an infinite beam resting on the ground surface (Vesic, 1961). Yu et al. (2013) and Attewell et al. (1986) both indicated that the tunnel (or pipeline)-soil relative stiffness exhibits a high sensitivity to the tunnel (or pipeline) embedment depth. In reality, shield tunnels are generally constructed in a certain depth below ground surface. Therefore, using the Vesic′s expression to estimate the tunnel-soil interaction associated with excavation may lead to misleading results. To overcome the shortcomings of previous semi-analytical methods, a simplified analytical method is proposed to evaluate the responses of existing shield tunnels to adjacent excavation. In this proposed method, the Pasternak foundation is used to simulate tunnel-ground interaction behaviors (Pasternak, 1954). A shear layer on the top of springs is introduced to consider the continuity of adjacent springs. It is expressed as

Fig. 1 shows that a rectangular basement is excavated above a existing shield tunnel. Fig. 2 shows the general relative position between the excavation and the existing shield tunnel. As shown in Fig. 1, the length, width and depth of the excavation are denoted by L, B and H, respectively. The centerline of underlying the existing shield tunnel is buried at the depth of h below the ground surface. The unloading stress imposing on the shield tunnel can be obtained using the Mindlin′s formula (Mindlin, 1936), ignoring the effects of the presence of the existing tunnel. Based on the Mindlin′s formula, the unloading vertical stress q(x) along the existing tunnel at the tunnel centerline level is obtained:

(2)

where Gc is the shear stiffness of the shear layer. Moreover, a modified coefficient of subgrade modulus expression, which is capable of considering the buried depth of tunnel, is also proposed. The proposed simplified method provides a rapid, effective and low cost estimation of a shield tunnel responses induced by an adjacent excavation engineering. The validity of the proposed method is examined by a three-dimensional finite element analysis and two published filed case histories. The results obtained from the proposed method are also compared and discussed with those from the Winklerbased method. Finally, parametric analyses are also carried out to investigate the effects of various factors on the deformation and internal forces of a shield tunnel, including the ground elastic modulus, excavation depth and geometry.

y o

Existing tunnel

L/2

Ș

d

L/2

Į

2. Analysis method

Excavation o Two-stage analysis method is selected and utilized in this analysis, which is widely used in structure-ground interaction problems (Zhang et al., 2013a; Zhang et al., 2013b; Zhang and Huang, 2014; Zhang et al., 2015; Yu et al., 2013). According to the two-stage analysis method, the excavation-tunnel interaction analysis is mainly divided into two individual but connected stages. First, the excavation induced vertical

x B/2

∂2w (x ) ∂x 2

2.1. The unloading stress caused by the adjacent excavation

Į

ȟ

B/2

p = kw (x )−Gc

unloading stress acting on the underlying tunnel is calculated by Mindlin′s Green function (Mindlin, 1936), ignoring the presence of existing shield tunnel. Second, the shield tunnel responses to the corresponding unloading stress are computed numerically using finite difference method.

Fig. 2. Plan view of the relative position between existing tunnel and excavation.

95

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q (x ) =

B

L

2

2

∫− B2 ∫− 2L

equation of a Euler-Bernoulli long beam resting on the Winkler model. Eq. (5) is a fourth-order homogenous difference equation and it is difficult to solve it analytically. In order to simplify the calculation process, the finite difference method is utilized to solve Eq. (5) numerically. Fig. 4 illustrates the discretization of a tunnel. In Fig. 4, the tunnel is divided into n + 5 elements of length l, including extra 2 virtual elements at the two ends of the tunnel, respectively. Correspondingly, Eq. (5) can be written in the finite difference form:

pdξ dη ⎡ (1−2ν )(z 0−H ) (1−2ν )(z 0−H ) − + 3 8π (1−ν ) ⎢ R R23 1 ⎣

−

3(z 0−H )3 3(3−4ν ) z 0 (z 0 + H )2−3h (z 0 + H )(5z 0−H ) − R15 R25

−

30hz 0 (z 0 + H )3 ⎤ ⎥ R27 ⎦

(3)

where ν is the Passion′s ratio; p is the total unloading vertical stress at n the excavation base, p = ∑i = 1 γi Hi , γi and Hi are the unite weight and thickness of soil at layer i, respectively . The parameters R1 and R2 are (ξ −l)2 + (η−b)2 + (z 0−H )2 and (ξ −l)2 + (η−b)2 + (z 0 + H )2 , respectively. Note that the parameters R1 and R2 control the relative position between the excavation and the shield tunnel. In order to compute the unloading stress along the existing shield tunnel more easily and directly, a new coordinate system x–y is applied in the analysis, as shown in Fig. 2. In the new coordinates, the origin point of x–y coordinates is placed at the centerline of the existing shield tunnel and the x-axis and the y-axis are parallel and perpendicular to the tunnel axis, respectively. The relationship between the new and the old coordinates can be obtained as follows:

⎧ ξ = x cosα + dcosα ⎨ ⎩ η = −x sinα + dsinα

(EI )eq

6wi−4(wi + 1 + wi − 1) + (wi + 2−wi − 2) w −2wi + wi − 1 + kDwi−Gc i + 1 l4 l2

= q (x i ) D

(6)

Eq. (6) can be further written in the matrix-vector form

[Kt ]{w} + [Ks]{w}−[G]{w} = {Q}

where [Kt ], [Ks] and [G] represent the stiffness matrixes of the bending beam element, foundation and shear layer, respectively. {w} and {Q} are the longitudinal shield tunnel displacement and unloading stress vectors, respectively. Note that {w} = {w0,w1,⋯wi,wi + 1 ⋯wn − 1,wn}T and {Q} = {q (x 0),q (x1) ⋯q (x i ) ⋯q (x n − 1),q (x n )}TD . {Q} can be obtained from Eq. (3). The stiffness matrix of foundation [Ks] can be expressed as

(4)

0⎤ ⎡1 ⎥ ⎢ 1 1 ⎥ ⎢ ⋱ ⎥ ⎢ ⎥ ⎢ [Ks] = Dk . 1 ⎥ ⎢ ⋱ ⎥ ⎢ 1 ⎥ ⎢ 1 ⎥ ⎢ 0 1 ⎦(n + 1) × (n + 1) ⎣

where d is the distance between the two origin points of the two coordinate systems; and the α is the angle between the x and ξ axes. Combining Eqs. (3) and (4), the unloading vertical stresses acting along the tunnel can be calculated rapidly. 2.2. Shield tunnel responses to unloading stress

(8)

Both two ends of tunnel are supposed to satisfy the free boundary condition. Therefore, the shear forces Q and bending moments M at the two ends are not induced:

Fig. 3 shows the excavation-shield tunnel calculation model. The shield tunnel is treated as a continuous Euler-Bernoulli long beam resting on the Pasternak foundation model. It is assumed that the tunnel is always in contact with the surrounding soils and the slippage between the tunnel-soil interface is not involved in the method. Moreover, the ground is assumed to be isotropic elastic material and the plastic behavior of the soil is not taken into account. The governing equilibrium equation for the deflection of a shield tunnel resting on the Pasternak foundation subjected to the unloading stress q(x) is provided below:

2

⎧ M0 = Mn = −(EI )eq d w (2x ) = 0 dx

⎨Q = Q = −(EI ) d3w (x ) = 0 0 n eq dx 3 ⎩

(9)

Similarly Eq. (9) can be also written in the finite difference form 2

⎧ M0 = −(EI )eq d w (2x ) = −(EI )eq w1 − 2w20 + w−1 = 0 dx

d4w (x ) d2w (x ) (EI )eq + kDw (x )−Gc D = q (x ) D dx 4 dx 2

(7)

l

⎨ M = −(EI ) d2w (x ) = −(EI ) wn + 1 − 2wn + wn − 1 = 0 n eq dx 2 eq l2 ⎩

(5)

where the (EI )eq is the equivalent bending stiffness of the shield tunnel; the w(x) is the deflection of the shield tunnel; and the D is the outer diameter of the shield tunnel. Note if the parameter of shear layer Gc is set to zero, the governing equation degenerates into the widely-used

, (10)

3

⎧Q0 = −(EI )eq d w (3x ) = −(EI )eq w2 − 2w1 + 23w−1 − w−2 = 0 dx

2l

⎨Q = −(EI ) d3w (x ) = −(EI ) wn + 2 − 2wn + 1 + 2wn − 1 − wn − 2 = 0 n eq dx 3 eq 2l3 ⎩

Ground surface

(11)

Fig. 3. The sketch of shield tunnel supported on Pasternak model subjected to excavation-induced unloading stress.

h

H

Excavation

Unloading stress q(x)

D

Tunnel Shear layer Gc Winkler spring k

96

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Fig. 4. Discretization of tunnel.

Tunnel

-1

-2

0

2

1

i-2

i-1

i

i+1 i+2

n-2

n-1

equilibrium, the equivalent bending stiffness of a shield tunnel is deduced by Shiba et al. (1988):

Combining Eqs. (10) and (11), the deflection of virtual points w−1, w−2, wn+1 and wn+2 at the two ends of shield tunnel can be obtained, respectively:

(EI )eq =

⎧ w−2 = 4w0−4w1 + w2 ⎪ w−1 = 2w0−w1 ⎨ wn + 1 = 2wn−wn − 1 ⎪ wn + 2 = 4wn−4wn − 1 + wn − 2 ⎩

cos3 ψ Ec Ic cosψ + (ψ + π /2)sinφ

(17b)

kb = Ej Aj / l j

(17c)

(13) and

Δ = φ (r + m) = 0 0⎤ 1 ⎥ ⋱ ⋱ ⎥ 1 −2 1 ⎥ 1 − 2 1⎥ 0 0 0⎥ ⎦(n + 1) × (n + 1)

(14)

l2

(15)

[KM ]{w }

where[KM ] is bending moment matrix and numerically[KM ] =

(18)

2.3.2. Determination of foundation parameters Many researchers have been presented various expressions to estimate the coefficient of subgrade modulus k (Biot, 1937; Terzaghi, 1955; Vesic, 1961). By further modifying Boit′s expression (Biot, 1937), Vesic (1961) derived a solution of coefficient of subgrade modulus kvesic by allowing an infinite beam resting on an isotropic elastic solid surface

where [K ]−1 is the inverse matrix of [K ]. Finally, the longitudinal shield tunnel displacement {w} caused by the excavation-induced unloading stress can be obtained numerically. Furthermore, combined with longitudinal shield tunnel displacement {w} , the bending moment {M } along the tunnel can be also obtained

(EI )eq

M (r + r sinψ) lb (EI )eq

where m is the distance from the tunnel centerline to the neutral axis, as shown in Fig. 5; r is the distance from the tunnel centerline to the bolt location, as shown in Fig. 5. M can be obtained from Eq. (16).

Supposing that [K ] = [Kt ] + [Ks]−[G], Eq. (7) can be solved

{w} = [K ]−1 {Q}

⎟

where kb is the elastic stiffness of longitudinal joints, Ej is the Young′s modulus of bolt; Aj is the section area of bolt; lj is the length of bolt; n is the number of longitudinal bolts; l is the width of a tunnel segment; Ec is the Young′s modulus of tunnel segment; ψ is the angle of neutral axis, as shown in Fig. 5; Ic is the longitudinal inertia moment of the section of a segment; and Ac is the sectional area of tunnel segments. When subjected to a bending moment M, opending of jointsΔ occurs and its maximum value locates at the edge of the tension side of the segmental ring, as shown in Fig. 5. Accorroding to the geometric relationship, the maximum opening of joints Δ can be expressed:

0 ⎤ ⎡ 2 −4 2 ⎥ ⎢− 2 5 − 4 1 ⎥ ⎢ 1 −4 6 −4 1 ⎥ 1 −4 6 −4 1 (EI )eq ⎢ ⎥ ⎢ ⋱ ⋱ ⋱ ⋱ ⋱ [Kt ] = 4 ⎥ ⎢ l 1 −4 6 −4 1 ⎥ ⎢ 1 −4 6 −4 1 ⎥ ⎢ 1 − 4 5 − 2⎥ ⎢ 2 − 4 2 ⎦(n + 1) × (n + 1) ⎣ 0

⎡0 0 ⎢1 − 2 ⋱ [G] = Gc D ⎢ ⎢ ⎢ ⎢0 ⎣

(17a)

nkb l ⎞ ψ + cotψ = π ⎛0.5 + E c Ac ⎠ ⎝ ⎜

(12)

Subsequently the bending beam element stiffness matrix [Kt ] and the shear layer stiffness matrix [G] can be obtained, respectively:

{M } = −

n n+1 n+2

kVesic =

0.65Es 12 Es B 4 B (1−ν 2) EI

(19)

where B is the width of the beam, EI is the bending stiffness of beam and Es is the elastic modulus of soils. It is found that the Vesic′s expression is obtained by allowing the beam resting on the ground surface. However, a shield tunnel is commonly buried at a certain depth under the ground surface and the tunnel-ground relative stiffness is highly sensitive to the tunnel embedment (Yu et al., 2013). Thus, the use of the subgrade modulus (Vesic′s expression) obtained from ground surface may likely lead to misleading results of tunnel (or pipeline) responses associated with external construction (Attewell et al., 1986; Klar et al., 2005). Attewell et al. (1986) suggested taking twice the value of the Vesic′s expression for a pipeline or a tunnel buried at infinite depth. It is expressed as

(16) 1 [G ]. Gc D

2.3. Determination of parameters 2.3.1. Determination of equivalent bending stiffness of shield tunnel In the practice, a shield tunnel segmental lining consists of various segmental pieces connected together by steel bolts. Due to the existence of joints, the bending stiffness of the shield tunnel in the longitudinal direction is significantly reduced compared to a continuous tunnel lining structure. Various researchers have proposed different analytical methods to evaluate the shield tunnel equivalent bending stiffness (Shiba et al., 1988; Liao et al., 2008; Ye et al., 2011). Among these current methods, Shiba′s method is popular and widely accepted by engineers. Fig. 5 shows the calculation model for Shiba′s method. According to the Shiba′s theory, the connected bolt and segmental lining withstands the tensile and compressive forces, respectively. When subjected to a positive bending moment, as shown in Fig. 5, the upper and the lower parts of segmental lining suffer tension and compression, respectively. Based on the deformation coordination and physical

k∞ = 2kVesic

(20)

where k∞ is the coefficient of subgrade modulus for a beam buried at infinite depth below ground surface. However, for urban metro tunnels in soft cities, like Hangzhou and Shanghai, shield tunnels are mainly buried at shallow depths varying from 10 m to 30 m below the ground surface. Thus, using the Attewll′s expression may lead to overestimation of the soil-tunnel stiffness, resulting in underestimation of opening of joints and internal forces of a shield tunnel. Yu et al. (2013) further considered the effect of beam 97

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Fig. 5. Deformation and stress of the lining (Shiba et al., 1988). Fig. 6. 3D finite element analysis model. (a) The three-dimensional finite element mesh adopted in this study, (b) Relative position between excavation and the existing shield tunnel.

(a) The three-dimensional finite element mesh adopted in this study

(b) Relative position between excavation and the existing shield tunnel modulus kh can be obtained

embedment depth on the subgrade modulus by introducing parameter η:

η=

kh =

k∞ kh

with η = 1 +

(21)

1 1.7h/ B

if

h/B > 0.5

k∞ 1.3Es 12 Es B 4 = η ηB (1−ν 2) EI

(23)

In this analysis B = D and EI = (EI)eq . Note that Eq. (23) is only suitable for the condition of h/B > 0.5. In practice, a shield tunnel burial depth easily satisfies this condition. The parameter of shear layer Gc is a crucial parameter in the Pasternak two-parameter model. Various methods have been proposed to evaluate the values of Gc (Kerr, 1985; Tanahashi, 2004). Tanahashi

(22)

where kh is the coefficient of subgrade modulus when the beam is buried at the depth of h. Combining Eqs. (19), (20) with (21), the coefficient of subgrade 98

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Tunnel displacement w (mm)

R. Liang et al.

(2004) suggested an empirical formula to estimate the value of Gc:

Gc =

Es ht 6(1 + ν )

(24)

where ht is the thickness of the shear layer and ht = 2.5D, as suggested by Xu (2005). 3. Verification

3 FEM Winkler-based method Proposed method

2

1

Heave

At first, a three-dimensional (3D) finite element analysis for the problem of shield tunnel responses to excavation is performed to examine the validity of the proposed simplified analytical method. The results calculated by the 3D finite element model are analyzed and compared with those from the proposed and Winkler-based methods. Note that the Winkler-based method is based on the Winkler foundation model and the coefficient of subgrade modulus k is estimated using Vesic′ expression (Eq. (19)), which corresponds to current semi-analytical methods (Zhang et al., 2013a; Zhang et al., 2013b, 2015). Then, field measurements from two published field cases are also analyzed and compared with the predictions computed by the proposed and Winkler-based methods, respectively.

0 Settlement -1 -100

-80

-60

-40

-20

0

20

40

60

80

100

Distance from basement center, x (m) Fig. 7. Comparison of tunnel displacements.

large deviation can be attributed to the fact that the Winkler-based method with the Vesic′s subgrade modulus underestimates the tunnelground relative stiffness for a buried shield tunnel. The prediction from the proposed method is in consistent with the FEM results, though maximum tunnel heave is slightly smaller than that of the FEM. It is probably because the removal of excavated soil may result in local reduction of the tunnel-ground relative stiffness, which cannot be further considered in the proposed method. Fig. 8 shows the comparison of bending moments computed by the FEM and predictions from the Winkler-based method and the proposed method. Positive and negative values of bending moment correspond to tension and compression of segmental rings, respectively, as shown in Fig. 5. Compared with the FEM result, the Winkler-based method overestimates the maximum positive and negative bending moments by 83% and 63%, respectively. It is because that the Winkler-based method underestimates the tunnel-ground relative stiffness, which finally results in overestimation of the bending moment of the tunnel. Although the maximum bending moment predicted by the proposed method is slightly larger than the FEM result, a reasonable good agreement is observed between the FEM and proposed method results. Generally the proposed method offers a conservative and satisfactory prediction of the tunnel bending moment when subjected to aboveexcavation. Fig. 9 shows the comparison of maximum opening of joints. Note that the maximum opening of joints of the shield tunnel in the finite element model are indirectly acquired using Eq. (18). Comparing Fig. 8 and Fig. 9, a consistent trend between the maximum opening of joints and the bending moment is observed. It is indicated that the distribution of the opening of joints is mainly controlled by the bending moment. Similar to bending moments, the Winkler-based method gives a remarkable overestimation of the opening of joints when compared with the FEM result. The reason for the overestimation may be due to

3.1. Comparison with 3D finite element model (FEM) A three-dimensional (3D) finite element analysis (FEM) is carried out to investigate the excavation-tunnel interaction using Plaxis 3D software (Plaxis, 2001). Fig. 6a and b show the 3D element mesh and the excavation-tunnel relative position, respectively. A excavation pit is constructed directly over a existing shield tunnel, as shown in Fig. 6a. The length L, width B and excavated depth H of the excavation in the 3D model are 8 m, 8 m and 6 m, respectively. To eliminate the boundary effects, the dimensions of 200 m × 200 m × 40 m are used in the numerical model. Four diaphragm walls are installed to support the lateral earth pressure and the depth of each diaphragm wall is 10 m. The ground is considered as isotropic elastic medium. The elastic modulus, Poisson′s ratio and unit weight of the soil are 15 MPa, 0.33 and 18 kN/m3, respectively. The shield tunnel is buried at the depth of 25 m below the ground surface. The shield tunnel is supposed to be a typical metro shield tunnel in Chinese soft areas and its related dimensions are summarized in Table 1. Based on the Shiba′s theory (Eq. (17)), the calculated equivalent bending stiffness (EI)eq is 7.8 × 104 MN m2. The shield tunnel is simulated using continuous elastic solid body. Due to the existence of segmental joints, the bending stiffness is significantly smaller than that of continuous of concrete tunnel structure. To consider the reduction in tunnel bending stiffness, the elastic of tunnel is set to be 2.825 GPa. Accordingly the bending stiffness of the shield tunnel in the 3D model is identical to the value computed using the Shiba′s theory. Fig. 7 shows the comparison of tunnel displacements computed by the FEM and predictions from the Winkler-based method and the proposed method. Compared with the FEM result, the Winkler-based method remarkably overestimates the tunnel heave. The reason for this Table 1 Tunnel lining parameters. Project

External diameter, Dt (mm)

Internal diameter, Di (mm)

Lining thickness, t (mm)

Lining width, ls (m)

Young′s modulus, Ec (MPa)

Metro tunnel Yan′an road tunnel

6200 11,000

5500 9900

350 550

1.2 1

3.45 × 104 3.45 × 104

Project

Number of longitudinal joint, N

Diameter of joint, Db(mm)

Length of joint, lb (mm)

Young′s modulus of joint, Eb (kPa)

(MN m2)

17 32

30 38

400 760

2.06 × 108 2.06 × 108

7.8 × 104 3.99 × 105

Metro tunnel Yan′an road tunnel

99

Equivalent bending stiffness, (EI )eq

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FEM Winkler-based method Proposed method

800

Bending moment M (kN m)

Tunnel displacement w(mm)

1000

600 400 200

18

Winkler-based method Proposed method Numerical analysis (Huang et al., 2012) Measured (Huang et al., 2012)

16 14 12 10 8 6

4 Heave 2

0

0

-200 -400 -100

Excavation width

-2 Settlement-80

-80

-60

-40

-20

0

20

40

60

80

-60

-40

-20

0

100

Bending moment M (kN m)

-2

80

Winkler-based method Proposed method

20000

8 FEM Winkler-based method Proposed method

6

60

25000

Fig. 8. Comparison of tunnel bending moments.

( 10 mm)

40

(a)

Distance from the excavation center, x (m)

Maximum opening of joints

20

Distance from the excavation center, x (m)

4 2

15000 10000 5000 0 -5000

0

Excavation width -10000

-2

-80

-60

-40

-20

0

20

40

60

80

Distance from excavation center, x (m) -4 -100

(b) -80

-60

-40

-20

0

20

40

60

80

100

0.4

(mm)

Distance from excavation center, x (m) Fig. 9. Comparison of maximum opening of joints.

Maximum opening of joints

underestimation of the soil-tunnel relative stiffness of the Winklerbased method. By further modifying the subgrade modulus and introducing the Pasternak foundation model, the proposed method generally gives a reasonable prediction of the shield tunnel opening of joints compared to the FEM result. From above analysis, the feasibility of the proposed method is eventually verified and it appears that the proposed method is a rapid and robust method offering an upper approximation of tunnel responses associated with an adjacent excavation.

Winkler-based method Proposed method

0.2

0.0

Excavation width

-0.2

3.2. Comparison with Bund Underground Passage project

-80

-60

-40

-20

0

20

40

60

80

Distance from the excavation center, x (m)

A section of Shanghai Bund Underground Passage using cut-andcover construction method was constructed over the northbound East Yan’an Road tunnel to facilitate the traffic condition in Shanghai, China. The existing East Yan′an Road tunnel was constructed using shield tunnelling technology and precast segmental rings were installed during shield tunnelling as permanent tunnel linings. The outer and inner diameters of the road tunnel are 11 m and 9.9 m, respectively. The lining thickness is 55 cm. Detailed dimensions of tunnel linings are summarized in Table 1. According to the Shiba′s theory, the equivalent bending stiffness of the road tunnel is approximate 3.99 × 105 MN m2. The skew angle between the Underground Passage and the existing tunnel alignments are 75°. The final excavation depth and the width of the Underground Passage are 11 m and 10 m, respectively. The

(c) Fig. 10. Comparison between computed and measured or numerical analysis results (a) comparison of tunnel displacement, (b) comparison of tunnel bending moment and (c) comparison of maximum opening of joints.

clearance between the Underground Passage base and the crown of underlying tunnels is 5.4 m. The excavation-induced unloading vertical pressure at the base of the passage excavation pit is about 189.3 kPa (Huang et al., 2014). Various countermeasures were carried out to alleviate the influence of excavation on the underlying existing tunnel. First, the excavation was separated into three sections by partition 100

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walls. Second, the ground improvement was preformed to increase the ground stiffness of soils around the tunnel. Furthermore, extra 50 kPa surcharge load was also imposed upon the base of the excavation to resist the heave of ground due to stress relief. Detailed information about the excavation procedure and subsoil condition can be found in Huang (2012) and Huang et al. (2013, 2014). Due to soil improvement, the average ground elastic modulus is taken as 30.8 MPa, according to Huang (2012). Fig. 10a shows the comparison of tunnel vertical displacement from the field measurement, the numerical analysis, the Winkler-based method and the proposed method. It appears that the predicted result by the Winkler-based method markedly gives an overestimation of the shield tunnel displacement compared with numerical analysis result and the measurement. The discrepancy between the Winkler-based method and the measurement or numerical analysis may be ascribed to the underestimation of tunnel-ground stiffness using the Vesic's expression. Thus, this method may be not suitable to describe the responses of buried shield tunnel to above-excavation. The predicted result from the proposed method with a modified subgrade modulus is in general consistent with the numerical analysis result and the measurement, although the prediction is slightly larger than them. This is probably due to the fact that the proposed method cannot further consider the effects of partition walls on shield tunnel, which made full use of time-space effect of excavation and significantly reduced the tunnel heave. Generally the proposed method offers a conservative solution of shield tunnel displacement when subjected to over-excavation. Fig. 10b and c show the variations of calculated bending moment and maximum opening of joints along tunnel longitudinal direction, respectively. It is found that the maximum bending moment and opening of joints occur at the positions directly below the excavation center. It is indicated that these sections beneath the excavation may be the most dangerous parts and much easier to damage. Accordingly special attention should be paid to protect these positions of the shield tunnel. Overestimations of both tunnel bending moment and maximum opening of joints computed by the Winkler-based method are also observed when compared to the results of the proposed method. It implies that excessively conservative protective measures may be likely carried out if the Winkler-based method is used to evaluate the responses of a shield tunnel to above-excavation.

y 9m

Excavation 45

13 m

66°

x

9m

13 m

Existing tunnel

6.5 m

Excavation 1

2.76 m

2

6.2

12.36 m

(a)

m

Underlying tunnel

(b) Fig. 11. Relative position between existing shield tunnel and above-excavation: (a) plan view; (b) sectional view.

Table 2 Relative soil parameters.

3.3. Comparison with Dongfang Road Underground Crossing Project in Shanghai The Dongfang Road Underground Crossing Project, located at the Pudong New District in Shanghai, China, was planned and designed to cope with the urban traffic stress of the central business center (Chen, 2005). The underground crossing project was constructed using cutand-cover construction method. A section tunnel of Shanghai metro Line-2 locates directly beneath the excavation pit. The relative position between the excavation and the underlying shield tunnel is shown in Fig. 11. To simplify the study, the excavation is roughly considered as a rectangle with 26 m in length and 18 m in width. The relative soil parameters are summarized at Table 2. The skew angle between the tunnel alignment and the x-axis is 45°, as shown in Fig. 10a. The depth of the excavation pit is 6.5 m and the excavation base to the crown of the tunnel is 2.76 m. The metro tunnel was constructed by employing shield tunnelling technology and precast segmental rings connected by steel bolts were installed as permanent linings. The axis of the tunnel is buried at the depth of 12.36 m below the ground surface. The dimensions of the shield tunnel are identical to those of metro tunnel listed in Table 1. Since the Metro Line-2 tunnel is one of most heavily travelled lines in the Shanghai transit system, it is imperative that its safety and integrity must be strongly guaranteed during the excavation. Thus, especial protective measures and engineering controlling techniques, including deep soil improvement, separated excavation procedures and

Expression modulus, Es0.1–0.2/ MPa

Poisson′ s ratio

18.5 18.4 17.7 17.7

6.34 3.71 4.43

0.4 0.3 0.3

2.28 2.46

18.3 17.2

9.72 3.63

0.35 0.35

8.7 2.41 3.89 4.25

16.6 17.9 18.1 19.4

2.27 4.07 4.55 6.09

0.35 0.4 0.4 0.35

Soil layer

Thickness

Unit weight, γs kN/m3

➀ Fill ➁1 Silty clay ➁2 Silty clay ➂1 Muddy silty clay ➂2 Sandy silt ➂3 Muddy silty clay ④Muddy clay ⑤1 Clay ⑤2 Silty clay ⑥Silty clay

1.82 1.13 0.82 1.08

tunnel reinforcement, were applied to minimize the disturbance of the above-excavation. To consider the increase in ground stiffness due to ground improvement, the soil elastic modulus is assumed to be 20 MPa in this analysis. A comparison of the tunnel displacement between the predictions and measurements is shown in Fig. 12. From inspection of this figure, the magnitude of tunnel displacement is remarkably overestimated by the Winkler-based method compared with the 101

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Meausred (Chen., 2005) Winkler-based method Proposed method

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Maximum tunnel heave wmax (mm)

Tunnel displacement w (mm)

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20mmby Shanghai Municipal Standard (2010)

20 15 10

Heave 5

0

55 50

(EI)eq

5(EI)eq

45

10(EI)eq

50(EI)eq

40 35 30

20 mm by Shanghai Municipal Standard (2010)

25 20 15 10

Settlement

5

-5

-50

100(EI)eq

-40

-30

-20

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Ground elastic modulus, Es (MPa)

Distance from the excavation center, x (m)

(a)

Fig. 12. Comparison of the measured and computed tunnel heaves due to excavation.

Maximum bending moment Mmax(kN m)

1000000

measurement. According to Shanghai Municipal Standard (2010), the induced metro tunnel displacement due to adjacent construction activities should not be allowed to exceed the allowable limit of 20 mm. It can be seen that the calculated maximum tunnel heave by the Winklerbased method almost reaches 30 mm, which is obviously beyond the allowable limit set by Shanghai Municipal Standard (2010). It means that excessively conservative protective measures will be likely preformed according to the predictions from the Winkler-based method. A consistently good agreement is observed between the prediction from proposed method and the measurement, though the predicted results slightly overestimate the tunnel heave. The reason for this slight overestimation may be ascribed to fact that the countermeasures (i.e. tunnel stiffening, separated excavation procedures), which applied prior to excavation and would reduce adverse effects on underlying tunnel, are not involved in the proposed method. Nevertheless, the proposed method still offers a reasonable approximation for tunnel responses associated with adjacent excavation. From above discussion, the validity of the proposed method is finally verified by comparisons with the Finite Element analysis and two published case histories for the problem of excavation-induced shield tunnel responses. It is indicated that the proposed method adopted in the analysis provides a rapid and effective method for evaluating shield tunnel responses subjected to above-excavation in the preliminary design stage.

(EI)eq

5(EI)eq

10(EI)eq

50(EI)eq

100(EI)eq

100000

10000

1000

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Ground elastic modulus, Es (MPa)

(b)

Maximum opening of joints

(mm) max

1.2

4. Parametric analyses In this section, a series of parametric studies are carried out to gain a greater understanding of the effects of different factors on the responses of shield tunnel associated with adjacent excavation, including the ground elastic modulus, excavation depth and excavation geometry. For a direct comparison with the corresponding different factors, an assumed example is selected directly in this study. In general, an excavation pit is constructed directly over a underlying shield tunnel. The length L, width B and excavation depth H of the excavation pit are set to be 30 m, 30 m and 8 m, respectively. The centerline of the tunnel is buried at 20 m below the ground surface. The average soil unit weight and Poisson′s ratio are 18 kN/m3 and 0.33, respectively. The existing tunnel is considered as a typical metro shield tunnel with bending stiffness of 7.8 × 104 kN m2 and its dimensions are listed in Table 1.

1 mm by MOHURD (2013)

1.0 0.8

(EI)eq

5(EI)eq

10(EI)eq

50(EI)eq

100(EI)eq

0.6 0.4 0.2 0.0 0

5

10

15

20

25

30

35

40

45

50

55

60

65

Ground elastic modulus, Es (MPa)

(c) Fig. 13. Effects of elastic modulus and bending stiffness on tunnel deformation and bending moment (a) maximum tunnel displacement, (b) maximum bending moment and (c) maximum of opening of joints.

during excavation process. Fig. 13a shows the relationship between the maximum tunnel heave at the excavation center and the ground elastic modulus. Five tunnel bending stiffness cases are considered to investigate the effects of elastic modulus on the tunnel heave. The equivalent tunnel bending stiffness (EI)eq is 7.8 × 104 MN m2, which is a typical flexural rigidity of shield tunnel in soft areas in China. Correspondingly, 5(EI)eq, 10(EI)eq, 50(EI)eq

4.1. Influence of ground elastic modulus In order to reduce the adverse effects on the existing tunnel due to above-excavation, protective countermeasures, such as ground improvement and tunnel stiffening, are frequently employed in practice 102

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Maximum tunnel displacement wmax (mm)

and 100 (EI)eq are 5, 10, 50 and 100 times of the tunnel bending stiffness (EI)eq of a metro shield tunnel at this section, respectively. At each tunnel bending stiffness, the maximum heave induced at the excavation center decrease rapidly with an increase in the ground elastic modulus. By increasing the ground elastic modulus from 10 MPa to 60 MPa, the heave induced in the tunnel is reduced by up to 80–84%. This is because a stiffer ground provides more resistance to resist the upward displacement of a shield tunnel due to the vertical stress relief. It is also found that if the elastic modulus is less than 25 MPa, the tunnel heave exceeds the allowable limit of 20 mm by Shanghai Municipal Standard (2010) when the tunnel is not especially stiffened. On the contrary, if the elastic modulus is larger than 30 MPa, tunnel heaves in all cases are within the allowable limit. It is indicated that by improving the ground strength may be one of most effective ways to reduce the tunnel heave induced by adjacent excavation. In practice, ground grouting as well as soil-cement mixing are frequently used in ground improvement. As the tunnel stiffness is increased, induced tunnel heave decreases with tunnel stiffness, as shown in Fig. 13a. By increasing the tunnel bending stiffness 100 times, the tunnel heave at the excavation center is reduced by up to 17.1 mm when the elastic modulus is 10 MPa. It means that increasing tunnel stiffness will significant reduce tunnel heave when the ground is relative soft. By increasing the elastic modulus up to 60 MPa, the heave difference between non-stiffened and stiffened tunnel (100 times) is only 1.3 mm. Besides, as the ground elastic modulus is increased further, induced heaves of all five cases seem to converge together. It appears that the heave differences between the non-stiffened or stiffened tunnel are gradually reduced with increasing ground elastic modulus. It is indicated that when the ground elastic modulus is large enough, the contribution of ground improvement to tunnel heave reduction prevails over that of tunnel stiffening. At such circumstance, there is no need to further conduct tunnel stiffening simultaneously. Fig. 13b shows the relationship between the maximum tunnel bending moment and the ground elastic modulus in logarithmic scale. At each tunnel stiffness, induced bending moment decreases linearly with an increase of the ground elastic modulus in logarithmic scale. It appears that increasing ground stiffness will effectively alleviate the tunnel bending moment induced by excavation. It is also demonstrated that a stiffer tunnel suffers much larger bending moments than those of a relative flexible one. This implies that stiffer tunnel may be more vulnerable to crack than a flexible tunnel when the unloading pressure is significant large. Fig. 13c shows the variation of maximum opening of joints with the ground elastic modulus. It is observed that openings of joints decrease with increasing ground elastic modulus. It is because that higher ground strength provides much more resistance to resist tunnel bending. By further increasing the tunnel flexural rigidity, the openings of joints decrease in a much smoother way. According to the Technical Code for Protection Structures of Urban Rail Transit issued by MOHURD (2013), the allowable limit of opening of joints is 1 mm when a shield tunnel is subjected to external construction activities. It is observed that induced opening of joints of non-stiffened tunnel exceeds the allowable limit if the ground elastic modulus is less than 12 MPa. From inspection of this figure, increasing ground elastic modulus and tunnel stiffness will both effectively reduce opening of joints when subjected to excavation. Similar to Fig. 13a, by further increasing the ground elastic modulus, the openings of joints of all five different stiffness tunnels seem to converge together and the differences between stiffened or non-stiffened tunnel are gradually reduced. It is indicated that if the ground stiffness is greater enough, the induced opening of joints is mainly dominated by the ground stiffness and the contribution of tunnel stiffening is insignificant.

h=20 m h=25 m h=30 m

30

20 mm by Shanghai Municipal Standard (2010)

20

10

0 0

2

4

6

8

10

12

14

Excavation depth H (m)

(a) Maximum tunnel bending moment Mmax (kN m)

10000 h=20 m h=25 m h=30 m

1000

100 0

2

4

6

8

10

12

14

Excavation depth H (m)

(b)

1 mm by MOHURD (2013)

1.0

Maximum opening of joints

max

(mm)

1.2

h=20 m h=25 m h=30 m

0.8 0.6 0.4 0.2 0.0

0

2

4

6

8

10

12

14

Excavation depth H(m)

(c) Fig. 14. Effects of excavation depth and tunnel buried depth on tunnel deformation and bending moment (a) maximum tunnel displacement, (b) maximum bending moment and (c) maximum opening of joints.

center with excavation depth. Note that the ground elastic modulus is set to be 20 MPa in this analysis. Three tunnel buried depths are also considered to explore the excavation-induced effects on the existing tunnel. Due to the larger unloading pressure acting on tunnel crown, at each tunnel buried depth, induced tunnel heave increases linearly with an increase in excavation depth, as expected. As the tunnel buried depth is increased, the induced tunnel heave with excavation depth is

4.2. Influence of excavation depth Fig. 14a shows the variation of tunnel displacements at excavation 103

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Maximum tunnel displacement (mm)

significantly reduced. Moreover, the tunnel heave differences between the shallow and deep tunnels remarkably increase with increasing excavation depth. When the tunnel is buried at 20 m below the ground surface, heaves of tunnel exceed the allowable limit of 20 mm by Shanghai Municipal Standard (2010) if the excavation depth is larger than 7.5 m. By increasing the tunnel buried depth to 30 m, the induced tunnel heave does not exceed the allowable limit until the excavation depth is over 11.5 m. It appears that a shallow tunnel located at a shallow depth suffers larger unloading stress and the tunnel-stiffness is relative smaller compared with a deep tunnel. Correspondingly it is much more vulnerable to damage than a deep embedded tunnel. Fig. 14b shows the variation of maximum bending moment with excavation depth. At each tunnel depth, the induced maximum tunnel bending moment gradually increases when the excavation depth varies from 2 m to 12 m in logarithmic scale. At a given excavation depth, the induced bending moment of deeper tunnel is significantly smaller than that of shallower tunnel. The reasons for this difference may be included two factors: less excavation-induced unloading stress on a deep tunnel and relative large tunnel-ground stiffness of a deep tunnel. It is also interesting to find that trends of tunnel bending moment are generally parallel to each other in logarithmic scale. Fig. 14 c shows the variation of maximum opening of joints with excavation depth. It can be seen that maximum opening of joints is sensitive to excavation depth. Similar to the variation of tunnel heave, maximum opening of joints increases rapidly with an increase in excavation depth. At a given excavation depth, as tunnel buried depth is increased, induced opening of joints is reduced. Moreover, the opening of joints differences between the shallow and deep tunnels dramatically increase by varying excavation depth from 2 m to 12 m. It implies that when the clearance between the tunnel crown and excavation base is smaller, more effective protective measures are needed to ensure the safety and integrity of tunnel structures. Besides, in these three cases, maximum openings of joints are all within the allowable limit by MOHURD (2013).

B=1H : L=1H L=10H L=1H: B=5H B=15H

7 6

L=5H L=15H B=10H

5 4 3 2

Heave 1 0

Settlement -1

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

Noramilased distance from excavation center, x/H

(a) Maximum tunnel heave wmax (mm)

8

Wide excavation Long excavation 6

4

2

0 0

5

10

15

20

25

Normalised excavation depth or width, B/H or L/H

4.3. Influence of excavation geometry

(b)

Fig. 15 shows the relative position between the excavation and the existing tunnel. The centerline of the shield tunnel runs parallel to long sides of excavation and the tunnel is located directly beneath the excavation (d = 0), as shown in Fig. 15. Note that the ground elastic modulus Es, the tunnel centerline depth h and the excavated depth H are set to be 35 MPa, 20 m and 8 m, respectively. Effects of different excavation lengths L and widths B on the existing shield tunnel are analyzed and discussed. For clarity of nomenclature, the excavation with long length and excavation with wide width are termed “long excavation” and “wide excavation”, respectively. Fig. 16a shows variations of tunnel longitudinal displacement with different excavation geometries, including the changes of excavation length L and width B. Note that the distance from the excavation center is normalized with respect to the excavation depth H in this analysis. For a long excavation, when the excavation width B is fixed (B = 1H), a significant increase in both the magnitude of tunnel heave and the influencing zone of shield tunnel are observed by varying the excavation length from 1H to 10H. It is indicated that a long excavation affects not only the magnitude of

Fig. 16. Effects of excavation width and length on tunnel displacement (a) tunnel displacement and (b) maximum tunnel displacement.

tunnel heave, but also the influencing zone of tunnel. By further increasing the excavation length L up to 15H, the influencing zone of tunnel expands much wider, while no significant increase in the maximum tunnel heave is observed. For a wide excavation, when the excavation length L is fixed (L = 1H), the tunnel heave increases rapidly by increasing excavation width from 1H to 10H. However, an insignificant increase in influencing zone of tunnel is observed. It is indicated that the tunnel influencing zone is relative insensitive to excavation width in a wide excavation. By further increasing the excavation width up to 15H, the incremental tunnel heave is almost negligible. At each identical excavation area, both the magnitude of tunnel heave and the influencing zone of tunnel in a long excavation are significantly larger than those of a wide excavation. Fig. 16b shows the variation of the maximum tunnel heave with normalized excavation depth L/H or width B/H. A clearly consistent trend of tunnel heave between wide and long excavation pits with excavation length or width is observed in the figure. By increasing the excavation width or length to 8 H, the maximum tunnel heave rapidly increases with the excavation width or length. Besides, at a given excavation area, the tunnel heave in a long excavation is always greater than that of a wide excavation and the heave difference between long and wide excavation increases with increasing excavation length or width. Moreover, less than 0.1% incremental tunnel heave is observed by further increasing the excavation width or length from 8H to 10H. Based on the above discussion, long side of excavation, if possible,

B

Existing tunnel Excavation

L Fig. 15. Relative position between excavation and existing tunnel.

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Biot, M.A., 1937. Bending of an infinite beam on an elastic foundation. J. Appl. Mech. 1 (4), 1–7. Burford, D., 1988. Heave of tunnels beneath the shell center, London, 1959–1986. Géotechnique 38 (1), 135–137. Chang, C., Sun, C., Duann, S.W., Hwang, R.N., 2001. Response of a Taipei Rapid Transit System (TRTS) tunnel to adjacent excavation. Tunnel. Undergr. Space Technol. 16 (3), 151–158. Chen, Y., 2005. Research on the heave displacement of tunnel induced by foundation pit. M.S. thesis Tongji University (in Chinese). Devriendt, M., Doughty, L., Morrison, P., Pillai, A., 2010. Displacement of tunnels from a basement excavation in London. Proc. ICE-Geotech. Eng. 163 (3), 131–145. Dolezalova, M., 2001. Tunnel complex unloaded by a deep excavation. Comput. Geotech. 28 (6), 469–493. Hu, Z.F., Yue, Z.Q., Zhou, J., Tham, L.G., 2003. Design and construction of a deep excavation in soft soils adjacent to the Shanghai Metro tunnels. Can. Geotech. J. 40 (5), 933–948. 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should be designed to be perpendicular to the existing tunnel longitudinal direction to reduce the potential adverse effects on the existing tunnel. Similar findings are also given by Shi et al. (2015) who explored the excavation-tunnel interaction using a 3D finite element analysis simulation. 5. Conclusions A simplified analytical method for evaluating shield tunnel responses due to adjacent excavation is proposed in the paper. The following conclusions can be drawn: (a) In the proposed method, the shield tunnel is treated as a continuous Euler-Bernoulli beam resting on the Pasternak foundation. A modified coefficient of subgrade modulus expression is presented to consider the effect of tunnel buried depth on the tunnel-ground stiffness. Two-stage analysis method is applied to analyze the problem of excavation-tunnel interaction. (b) The feasibility of the proposed method is validated by a 3D finite element analysis and two published measurements. The predictions from the proposed method are generally consistent with the results from the 3D finite element analysis and field measurements. However, the Winkler-based method usually overestimates the responses of tunnel when subjected to adjacent excavation. In general, the proposed simplified method provides a rapid and effective approach for evaluating the existing shield tunnel responses due to adjacent excavation. (c) Parametric analyses are also performed to investigate the effects of different factors on tunnel responses, including ground elastic modulus, excavation depth and excavation geometry. (d) Increasing ground elastic modulus will significantly reduce adverse effects on the shield tunnel. It is indicated that ground improvement will be an effective way to protect tunnel from adverse effects induced by adjacent excavation. (e) Tunnel deformation and bending moment are both sensitive to excavation depth. Close tunnel-excavation base clearance will induce large tunnel heave, opening of joints and bending moment. (f) Increasing excavation length and width will both increase tunnel heave. Long excavation affects not only the magnitude of tunnel heave, but also the influencing zone of tunnel. Thus, the adverse effects induced by a long excavation are more serious than a wide excavation. (g) The proposed method is especially suitable for the circumstance where a existing tunnel locates directly beneath excavation pit. It should be noted that the proposed simplified analytical method does not consider the nonlinearity of tunnel-ground interaction. The effects of supported system and layered property of ground are not included in the study. These shortcomings should be further considered in the future. Acknowledgments The authors gratefully acknowledge the financial support provided by National Natural Science Foundation of China (No. 41472284, No. 51309207, No. 51678547 and No. 41502304), China Postdoctoral Science Foundation (No. 2015M581940 and No. 2017T100664), Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (No. CUG170647) and Ministry of Housing and Urban-Rural Development of China (No. 2014-K3-026). The corresponding author also gratefully acknowledges the support from my wife Ms. Zhuo Feng. References Attewell, P.B., Yeates, J., Selby, A.R., 1986. Soil Movements Induced by Tunnelling and Their Effects on Pipelines and Structures. Blackie and Son Ltd., London.

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