Analytical prediction of tunneling-induced ground movements and liner deformation in saturated soils considering influences of shield air pressure

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Analytical prediction of tunneling-induced ground movements and liner deformation in saturated soils considering influences of shield air pressure Zhiguo Zhang Associate Professor , Maosong Huang Professor , Chengping Zhang Professor , Kangming Jiang Postgraduate , Qiaomu Bai Postgraduate PII: DOI: Reference:

S0307-904X(19)30613-4 https://doi.org/10.1016/j.apm.2019.10.025 APM 13086

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

12 April 2019 1 October 2019 8 October 2019

Please cite this article as: Zhiguo Zhang Associate Professor , Maosong Huang Professor , Chengping Zhang Professor , Kangming Jiang Postgraduate , Qiaomu Bai Postgraduate , Analytical prediction of tunneling-induced ground movements and liner deformation in saturated soils considering influences of shield air pressure, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.10.025

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1 Highlights 

An analytical solution is presented for predicting long- and short-term ground deformation and liner internal forces.



Shield excavation effects with and without air pressure are both considered in saturated soils.



Oval-shaped convergence pattern is incorporated as deformation boundary condition.



Parametric analysis is conducted to measure circumference rules for liner displacements and internal forces.

2

Analytical prediction of tunneling-induced ground movements and liner deformation in saturated soils considering influences of shield air pressure Zhiguo Zhanga, Maosong Huangb, Chengping Zhangc, Kangming Jiangd, Qiaomu Baie a

Associate Professor, Postdoctor, School of Environment and Architecture, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China;

Department of Civil and Environmental Engineering, National University of Singapore, Singapore; Key Laboratory of Urban Underground Engineering of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China. (Corresponding author). E-mail: [email protected]. b

c

Professor, Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China. Professor, Key Laboratory of Urban Underground Engineering of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China.

d

Postgraduate, School of Environment and Architecture, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China.

e

Postgraduate, School of Environment and Architecture, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, China.

ABSTRACT: Complex underground constructions in urban areas require strict predictions for ground movements and liner deformation induced by shield-driven tunneling, in which the complex interaction mechanics between ground and liner play a substantial role. Previous studies, however, provided little information on the ground-liner interaction and less attention to the effects of groundwater and compressed air during the shield operation. This paper presents a closed-form analytical solution for predicting long- and short-term ground deformation and liner internal forces induced by tunneling in saturated soils in which shield excavation

3 effects with and without air pressure are both considered. The oval-shaped convergence deformation pattern is incorporated as the boundary condition of displacements around the tunnel section. This paper also investigates the difference between uniform radial and oval-shaped convergence deformation patterns on the ground and tunnel responses. Generally, the predicted ground movements by the oval-shaped deformation pattern aligns well with measured data of actual tunnels with and without considering the shield air pressure. It is comparatively observed that the shield excavation under air pressure obtains larger ground deformation than the non-pressure condition, and the long-term ground settlements induced by tunneling in saturated soils are confidently larger than the short-term. Moreover, the effects of sensitive parameters, including the shield air pressure, the long- and short-term effects on the tunneling-induced ground movements are assessed based on the oval-shaped deformation pattern. Furthermore, parametric analyses are conducted to measure the influences of concerned tunneling coefficients on the liner displacements and internal forces, namely, soil Young’s modulus, soil unit weight, coefficient of lateral soil pressure, tunnel radius, tunnel buried depth and gap parameter. In summary, the analytical approach proposed in this research provides an effective insight into the ground-liner interaction mechanics related with the shield air pressure, which can serve as an alternative approach in the preliminary design for conservatively estimating the excavation influences caused by tunneling in saturated soils. KEYWORDS: shield-driven tunneling; air pressure; ground-liner interaction; saturated soils; oval-shaped deformation pattern; ground movements

4 1. Introduction Owing to the fast-urban developments and the rapid growth of urban population, the underground public facilities are highly demanded in cities. Shield tunneling has become more widely used in subway constructions but the underground constructions inevitably causing the ground to deform [1-8]. It is unavoidable that the neighboring geo-environments, such as the surface buildings and buries structures, will be subject to excessive differential settlements due to the tunnel excavation [9-19]. Such tunnels are often buried in shallow depth where the ground tends to be saturated near the groundwater table, especially for the coastal region and river basin. As the leading factors of ground deformation induced by shallow tunneling in saturated soils, few studies took account of the effects of groundwater and air pressure generated by the working compressed air on the ground movements and liner internal forces. Therefore, it is of primarily theoretical and practical significance to perform a detailed study on related mechanical problems of shallow tunnels in saturated soils. Regarding

the

recent

attempts

to

develop

theoretical

solutions

for

tunneling-induced ground movements, the methods are commonly classified into three categories: empirical [20-27], numerical [28-34] (part for tunnel-soil-structures interaction) and analytical [35-57] methods. Empirical methods are still widely used on this subject. Based on the mass data of surface settlements, Peck [20] and Schmidt [21] approximated the ground settlement trough by the Gaussian distribution curve and proposed the formulas to estimate lateral distribution of ground settlements, which are also widely accepted in engineering practice [22,23]. Subsequently,

5 considerable investigators [24-27] adjusted the correlation coefficients of the Gaussian distribution curve [20,21] for better applicability and serviceability of tunnels. Nevertheless, the empirical methods are closely tied to measured data and intuitive deductions, which lead to lack of theory and application problems. Therefore, many investigators [28-34] selected numerical tools for soil behavior analyses. Powerful numerical methods appropriately provide full-scale simulations for specific construction processes and conditions of tunnels. Though the numerical methods are extremely helpful, the accuracy of the prediction heavily relies on models of practical conditions and soil behavior, which are often ideally assumed. In terms of analytical and semi-analytical methods, considerable researchers [35-57] adopted the physical concept of gap parameters by Lee et al. [58] and attained relevant estimating formulas through explicit theoretical derivations. Typical approaches include the point source theory (i.e., virtual image technique) [35,36,38], the complex variable method [37,39,42,46,49-51,54-57], the stochastic medium theory [44,47,52], and the stress function elastic method [40,41,43,45]. In particular, the force controlled method (FCM) and the displacement controlled method (DCM) are two leading approaches associated with tunneling. Numerical simulation applies forces corresponding to a fraction of the initial stress-state, to the nodes on the tunnel boundary. It can consider the nonlinear interaction between the existing structure and its surrounding soil, the soil elastoplastic behavior, the freedom for the tunnel to heave, and the complexity of construction operations. However, even though the ground loss is the main reason for the deformation of surrounding soils

6 affected by tunneling, but arbitrary ground loss cannot be controlled in FCM. In order to simplify the complex tunneling process and reflect the arbitrary ground loss, the displacement controlled method is favored because the effect of tunneling is simulated by adding displacements to nodes around the tunnel rather than by adding forces, in which the displacement boundary condition is imposed at the periphery of tunnel opening to simulate the stress release process induced by tunneling. Thus, the tunneling-induced deformation of surrounding soils can be directly discussed. Attempting to describe the effects of tunnel excavation, most researchers selected the displacement controlled method for closed-form analyses. Though this method cannot fully simulate the entire process of tunnel excavation, it can appropriately reflect the soil loss due to the physical clearance of tunneling machine tailskin. Many researchers [35-37,39-41,44,46,47,52-54,56,57] introduced the boundary conditions for uniform deformation at the periphery of tunnel opening (Fig. 1a). Furthermore, Loganathan and Poulos [38] presented a modified solution from Verruijt and Booker [36] and employed the boundary conditions for the non-uniform deformation that took into account the long-term effects of ovalization deformation of tunnel liner (Fig. 1b). According to Loganathan and Poulos [38], when the portion of the soil above the tunnel crown touches the tunnel liner, the soil at the side of the tunnel displaces towards the bottom of the tunnel, resulting into limited upward movement of the soil below the tunnel. When the tunnel liner settles on the bottom of the annulus gap (due to the self-weight) the distance between the crown of the tunnel liner and the crown of the excavated surface becomes twice the thickness of the annulus gap. Park [43]

7 introduced four trigonometric functional deformation patterns at the periphery of tunnel opening and particularly determined the deformation parameters of the excavation boundary (Fig. 2), which were widely recognized by various investigators [46,50,51,53,55]. Furthermore, the BC-4 deformation pattern [43] fully considers the opening ovalization of long-term deformation, which is delightfully similar to the deformation pattern proposed by Loganathan and Poulos [38]. In fact, few of existing analytical studies have investigated the effects of saturated soils and air pressure generated by the compressed air during shield tunneling, especially for the long-term effects of ground deformation and liner stresses. Timoshenko and Goodier [59] proposed a general solution in the form of the Airy function for ground displacement based on the elasticity theory. In the study of the stress function method in polar coordinate system, Bobet [40] presented analytical solution for the long- and short -term effect of air pressure and water pressure on surface settlement in shallow tunnel with saturated ground. Based on Bobet [40], Chou and Bobet [41] validated the predicted solution for shallow tunnels with twenty-eight tunnels. They pioneered the long- and short-term analyses of the ground settlements due to tunneling and created a new insight for understanding the ground-liner interaction mechanism, which has remarkably inspired the author of this paper. However, the tunnel section was considered to follow the uniform radial convergence deformation pattern alone in their studies. Observations from centrifuge field cases that more crown settlement and less invert heave should be considered in comparison with uniform convergence pattern [38]. They did not account for the

8 ovalization deformation existing in the course of tunneling or give detailed analysis for the internal forces of the segment liner. An analytical solution was proposed by Park [43] for tunneling-induced deformation in the oval-shaped displacement pattern and the outstanding contribution for Park [43] was a quantitative approach to consider real non-uniform ground deformation excavation influences. Despite the oval-shaped deformation pattern, Park [43] did not reflect the interaction mechanics between the ground and liner or consider factors such as the groundwater and shield compressed air. Based on the above-mentioned researches, Zhang et al. [53,55] focused on the interaction mechanism between tunnel liner and surrounding soil and evaluated predictions from an analytical solution for ground deformation and liner internal forces for shallow tunnels in clays. The stress function analytical solution by Zhang et al. [53] contained physical drawbacks, which were obviously not satisfied by the common condition. Such as, the above solutions for a shallow tunnel predicted an increase of displacements with the logarithm of the radial distance. To overcome the above disadvantages, an analytical solution by the complex variable method was presented by Zhang et al. [55] for problems in evaluating the ground movements induced by tunneling in clays. However, Zhang et al. [53,55] did not consider the long- and short-term effects of the groundwater or excavation influences for shield compressed air, which may adversely affect the applicability and accuracy for practical projects [60-61]. Furthermore, the parametric analysis for liner displacement was also not given in their studies. Generally, existing solutions have been very limited for estimating either

9 theoretically or numerically the complex excavation environments either shield air pressure or saturated soils, but they have contributed to the understanding of the ground-liner interaction mechanisms. According to the deformation pattern in the ovalization displacement boundary condition BC-4 [43], this paper investigates effects of groundwater and compressed air based on the ovalization deformation at the periphery of tunnel opening and the interaction between the disturbed ground and segment liner. A closed-form analytical solution for the long- and short-term effects of ground and liner deformation for shallow tunnels in saturated soils is presented and verified by the measured data for applicability. In addition, an analytical solution is further deduced for the liner internal forces of the corresponding boundary surface, based on the liner stress-internal force relation. Parametric analyses are conducted for the distribution pattern of circumferential displacements and internal forces at the periphery of tunnel opening. 2. Analysis model for shallow tunnel with liner The long- and short-term effects of the tunneling-induced disturbance are usually researched separately, for they affect the construction differently. Subjected to plane strain conditions, the analysis model for shallow tunnels measures the dissipation time of excess pore pressures of soil for the distinction between the long- and short-term effects. It is generally considered that the short-term effects of the tunneling-induced disturbance exist from shield tunneling to liner assembling, for the time is short and the excess pore pressures are slow to dissipate; whereas the long-term effects exist after a period following the completion of the excavation engineering, in which the excess pore pressures gradually dissipate. Therefore, we need to study two above

10 situations separately. This model introduces water pressure generated in the course of construction and assumes the boundary surface of liner is completely undrained, which implies the confining pressure of tunnel is literally the pore water pressures at the ground-liner interface. During tunneling, the shield applies an internal air pressure pa to prevent groundwater inflow into the tunnel and stabilize the ground. Once the excavation is finished, the liner is installed and the air pressure released. The analysis model of shallow tunnel can be seen in Fig. 3, in which x, y are Cartesian coordinates with origin at the center; and r, θ are the corresponding polar coordinates. In this figure, h and hw are the depth of the tunnel below the ground surface and below the water table, respectively; Ur and Uθ are the radius and circumferential ground displacement in polar coordinates; σr, σθ, and τrθ are the radial stress, hoop stress and shear stress of soil in polar coordinates, respectively; the unit weight, Young’s modulus and Poisson's ratio of soil are γ, E and v, respectively; the Young’s modulus and Poisson's ratio of the liner are Es and vs; and t is the thickness of the liner. The basic assumptions are as follows: (a) the ground is saturated, and both the ground and liner are homogeneous, isotropic and elastic; (b) the tunneling surface is a circle with the radius of r0; (c) deformation difference exists at the ground-liner interface and is coordinated by the ground-liner interaction. Additionally, the boundary condition BC-4 proposed by Park [43] is imposed as the coordinated deformation pattern, which can be further expressed as:

Ur

r  r0

 U rs

r  r0

1   u0 (5  3sin   3cos 2  ) 4

(1)

11 where U r and U rs are the radial displacements of the ground and liner, respectively; r0 is the radius of the tunnel; u0 is the difference value in radius between the shield and the liner, and u0  0.5g , g is the gap parameter estimated by following the procedure introduced by Loganathan and Poulos [38]. The gap parameter “g” can be estimated as shown: g = Gp + U*3D + ω, where Gp is the physical gap between the liner and the perimeter of the excavation; U*3D is a measure of the soil movements ahead of the face of the tunnel; and ω is the workmanship, such as the overcutting and so on; (d) no friction or relative slippage exists between the ground and liner, thus the shear stress  at the ground-liner interface can be expressed as:



r  r0

0

(2)

3. Analytical solution for short-term effects 3.1. Short-term analytical solution for ground displacements and liner internal forces without considering air pressure According to the elasticity problem, Timoshenko and Goodier [59] proposed a general solution in the form of the Airy function, which can be expressed as:

1   a0 ln r  b0 r 2  c0 r 2 ln r  d 0 r 2  a0  a1r sin  2 1 (b1r 3  a1r 1  b1r ln r ) cos   c1r cos  2 

(d1r  c1r 1  d1r ln r ) sin    (an r n  bn r n  2  anr  n

(3)

3

n2



bnr  n  2 ) cos n   (cn r n  d n r n  2  cnr  n  d nr  n  2 ) sin n n2

where the parameters a0 , b0 , c0 , etc., are the pending constants determined from the boundary conditions. As seen in Fig. 3, the solution for the ground displacements and liner stresses

12 must satisfy the equilibrium equations, strain compatibility equations and boundary conditions. The complete solution is obtained by decoupling the liner and ground and imposing compatibility of displacements and stresses at the interface between the ground and liner. The boundary conditions can be obtained as follows:



  /2

U Ur

r  r0

 U rs

r  r0

  /2

(4)

0

1   u0 (5  3sin   3cos 2  ) 4

r r

0

r  r0

r  r0

  rs

  rs

r  r0

r  r0

0

(5) (6) (7) (8)

where  rs and  rs are the radial stress and shear stress of the liner, respectively. Compared with a deep tunnel, the free surface of a shallow tunnel significantly affects the stresses and displacements around the tunnel section. The boundary conditions of the far field stresses are as follows:

y

r 

 ( b   w )(h  r sin  )  ( b   w )(h  r sin  )

x

r 

 k y + w (hw  r sin  )

(9) (10)

where  y and  x are the vertical and horizontal stresses of the soil in Cartesian coordinates, respectively (   is for the effective stress);  b and  w are the buoyant unit weight of soil and the unit weight of water, respectively; and k is the lateral earth pressure coefficient at rest. Due to the assumption of the short-term analysis (i.e. the excess pore pressures do not dissipate), the ground cannot change volume. Hence, for the poroelastic medium foundation, the boundary condition is introduced:

13

 r     0

(11)

According to Flügge [62], the stress-displacement relations for the liner are listed below:

d 2Us dU rs C (1  2 ) s    r0 r d 2 d E

(12)

dUs d 2U rs C d 4U rs C (1  2 ) s  U rs  (  2  U )  r0 rs r d F d 4 d 2 E

(13)

where E and v are the Young’s modulus and Poisson’s ratio of soil, respectively; C and F are the compressibility and flexibility ratios of the liner, respectively, defined as:

Er0 (1  s 2 ) Es As (1  2 )

(14)

Er03 (1  s 2 ) F Es I s (1  2 )

(15)

C

where Es and vs are the Young’s modulus and Poisson’s ratio of the liner, respectively; As is the cross-sectional area of the liner per unit meter; Is is the moment of inertia for the center of the liner per unit meter. The solutions of soil stresses are derived as follows:

a0   b hk  6a2r 4  4b2r 2 2 r   b rk  2 c1 r 3  c1 r 1  36c3r 5  d1r 1  30d3r 3  sin      b h(1  k )  12a2r 4  8b2r 2  sin 2      b r (1  k )  48c3r 5  40d3r 3  sin 3   

r 

(16)

14

a0   b h  6a2r 4 2 r   b r  2 c1 r 3  36c 3r  5 d r1  1 6d r3 3sin      b h(1  k )  12a2r 4  sin 2      b r (1  k )  48c3r 5  8d3r 3  sin 3   

  

(17)

 r   b r (1  k )  2 c1 r 3  36c 3r  5 d r1  1 18d r3 3cos  



  b h(1  k )  12a2r 4  4b2r 2  sin  cos      b r (1  k )  48c3r 5  24d3r 3  cos3   

(18)

Due to the short-term effects of the construction in saturated soils, assumption is made that the Poisson's ratio of soil is known and given by the limit value (i.e. ν=0.5). Then the ground radial and circumferential displacements, U r and U , are attained:

Ur 

1   a0 1 3 1 2 4 2   2a2r  2b2r  [c1r  c1 ln r  9c3r  9d3r ]sin  E  r 2

[4a2r 3  4b2r 1 ]sin 2   [12c3r 4 12d3r 2 ]sin 3 

U 

1  E



  2 1 4 2 [c1 r  c1 (1  ln r )  9c3r  3d3r ]cos  2 

4a2r sin  cos  [12c3r  4d3r ]cos  3

4

2

3



(19)

(20)

Substituting Eqs. (4-15) into Eqs. (16-19), the pending parameters are determined as follows: 2 2 1 2[ b h 1  k   2 w hw ] 1   CFr0  E  C  F  u0 (5  3sin   3cos  ) a0  r0 4  C  F 1    1  2  CF

c1  (  b + w )r02 c1  

a2  

(21a) (21b)

1 1  k   b r04 8

(21c)

d1  0

(21d)

1 [1   F  3] b h 1  k  4 r0 4 1   F  6

(21e)

15

b2 

1  2 1   F  3  b h 1  k  2 r0 4 1   F  6

(21f)

1 1   F  12  b 1  k  r06 12 1   F  24

(21g)

1 1   F  8  b 1  k  r04 8 1   F  24

(21h)

c3 

d3  

It should be noted that a function “Inr” exists in Eqs. (19) and (20), which reflects a continuing increase of the displacement distributions of the shallow tunnel as the distance increases. Clearly, Eqs. (19) and (20) do not satisfy the practical condition U r

r 

 0 . Hence, this paper introduces the boundary of zero vertical

displacement and modifies expressions proposed by Park [43] (using -Uθ instead of Uθ in the estimation of surface settlements). Finally, compositing the ground radial and circumferential displacements in the polar coordinates, the short-term ground displacements in the Cartesian coordinates can be expressed as:

U y  U r sin   U cos  

1  v  ' 2 1 ' 4 ' 2 c1r  c1 1  ln r   3c3r  d 3r E  2

a  1    0  2b2' r 1  2a2' r 3  sin    c1 1  2 ln r   6c3' r 4  4c3' r 4  sin 2  2   r 4b2' r 1 sin 3   8d3' r 2 sin 4  U x  U r cos   U sin  

(22)

1  v   a0 ' 3 ' 1     6a2 r  2b2 r  cos  E  r 

1     2c1' r 2  c1  6d3' r 2  cos  sin  2   4  2a2' r 3  b2' r 1  cos3   8d3' r 2 cos3  sin 

(23)



The maximum vertical displacement occurs at r=h,    / 2 , which is given below:

 max 



1  1  (a0  2b2 )h1  (c1  3 d3 )h 2  2a2h 3 3c3h 4  c1 ln h  E 2 

(24)

16 According to Eqs. (21) and (24), the ground surface settlements are dramatically affected by the radial difference between the shield tailskin and liner. Close correlation is found between the sharp increases of the ground settlements near the excavation surface above the tunnel and the oval-shaped convergence deformation caused by the deviatoric stresses. Under the oval-shaped convergence deformation pattern, the elastic expressions for the internal forces of the liner per unit meter can be obtained based on the analytical solutions of the ground stress and displacement field and the ground-liner interaction conditions (i.e. the stress-internal force relation of the liner, the stress-displacement interaction between the ground and liner). As indicated by Eqs. (25) and (26), Flügge [62] proposed the expressions for the stress-internal force relation of the liner at the excavation surface:

r0

dT dM   r02 rs d d

r0T 

d 2M  r02 rs 2 d

(25) (26)

where T and M are the axial force and bending moment acting on the cross section of the liner per unit meter along the tunnel, respectively;  rs and  rs are the radial and shear stress of the liner. Combining Eqs. (25) and (26) with the boundary condition Eq. (8) and the radial soil stress expression Eq. (16), the elastic solutions for the axial force and bending moment are solved as derived as follows:

17 1 [ Eu0 (2  3sin   3sin 2  )  2 b hr0 (1  k )(1  )](C  F ) 4 (C  F )(1  )  (1  2 )CF 3 h(1  k ) 3   b 1  k  r02 sin   b r0 sin 2  (1  ) F  24 (1  ) F  6 4   b (1  k )r02 sin 3  (1  ) F  24 CF 3  b h(1  k )   w hw r0  r0 C  F  (1  v)CF 2 (1  ) F  6

(27)

3  b h(1  k ) 2 3 r0   b h(1  k )r03 sin  2 (1  ) F  6 (1  ) F  6 3 h(1  k ) 2 2 4  b r0 sin    b (1  k )r03 sin 3  (1  ) F  6 (1  ) F  6

(28)

T

M 

3.2. Short-term analytical solution for ground displacements and liner internal forces with considering air pressure During the tunneling with compressed air, once the liner is installed, the air pressure will be released. Then, the excess pore pressures gradually dissipate as the excavation is completed. Therefore, the tunneling-induced deformation of the ground and liner with the compressed air can be simplified into two steps (i.e., the excavation process and the liner installation process) to calculate separately. It should be emphasized that the above-mentioned excess pore pressures do not dissipate from the excavation to the liner installation due to the assumptions of the short-term effects. 3.2.1. First step analyses for excavation process In this process, as the excavation opening is formed, the shield head provides air pressure pa to prevent groundwater inflow into the tunnel. Therefore, the value of the air pressure pa is generally taken as the magnitude of the water pressure at the invert of the tunnel, which can be expressed:

18

pa   w (hw  r0 )

(29)

To simplify the problem, this study divides the analysis of the elastic deformation effects of the saturated soils into two parts: (1) a stage regardless of the groundwater and air pressure; (2) a stage considering the actions of the groundwater and air pressure only. Superpose the ground deformation effects in the above-mentioned stages after solving them separately. Regardless of the groundwater and air pressure, the problem can be viewed as an unlined excavation problem, which is similar to the study by Park [43]. The ground stresses and displacements are the same as Eqs. (16-20), whereas the corresponding pending parameters are obtained as follows:

1 a0   b h(1  k )r02 2

(30a)

c1   b r02

(30b)

c1'  

1 1  k   b r04 8

(30c)

d1'  0

(30d)

1 a2'    b h(1  k )r04 4 1 b2'   b h(1  k )r02 2 1 c3'   b 1  k  r06 12 1 d3'    b 1  k  r04 8

(30e) (30f) (30g) (30h)

For the stages considering the actions of the groundwater and air pressure only, the corresponding boundary conditions are introduced at the distal stress field and the excavation boundary surface:

r

r  r0

 u

r  r0

  pa

(31a)

19



r

r 

 

r  r0

0

(31b)

=  u   w (hw  r sin  )

(31c)



(31d)

r 

r 

0

where u is the pore pressure;  w is the unit weight of the groundwater; and hw is the depth below the groundwater table. Furthermore, considering the action of the groundwater and air pressure only, the solution for the ground stresses and displacements are as follows:

 r  ( w hw  pa )

r02 1 r02   w sin  r2 2 r

   ( w hw  pa )

r02 1 r02   w sin  r2 2 r

(32)

(33)

 0

(34)

1 r2 u   w 0 sin    w (hw  r sin  ) 2 r

(35)

Ur  

r2 1 1 v [( w hw  pa ) 0   w r02 ln r sin  ] E r 2

U 

1 v 1 [  w r02 (1  ln r ) cos  ] E 2

(36) (37)

where  r and    are the effective radial and circumferential stresses of soil, respectively. 3.2.2. Second step analyses for liner installation process Similar to the solution process of Section 3.1, this process can be viewed as a lined excavation problem. However, subjected to the compressed air, there is still an internal pressure of pa around the circumference of the tunnel. The new boundary conditions are as follows:

20

r

  rs0  pa

(38a)

  rs0 =0

(38b)

  pa   w (hw  r0 sin  )

(38c)

 u

r  r0

r  r0

r  r0

where  rs0 and  rs0 , respectively, are the liner radial stress and shear stress. Combining the boundary conditions (6), (12), (13) and (38) above, the solutions for the ground stresses and displacements are expressed as:

 r 

1 4(1  v 2 )CFpa r02  (C  F ) Eu0 (5  3sin   3cos 2  )r0 4r 2 (1  v)(C  F )  (1  v 2 )CF

(39)

   

1 4(1  v 2 )CFpa r02  (C  F ) Eu0 (5  3sin   3cos 2  )r0 4r 2 (1  v)(C  F )  (1  v 2 )CF

(40)

 0

(41)

u0

(42)

1+v 4(1  v 2 )CFpa r02  (C  F ) Eu0 (5  3sin   3cos 2  )r0 Ur   4 Er (1  v)(C  F )  (1  v 2 )CF

U  0

(43) (44)

The two-stage solutions for the ground deformation are completely shown above. Subjected to the air pressure, the full expressions for the saturated soil stress and displacements can be obtained by superposing the results gained in each stage. Similarly, combining Eqs. (25) and (26) with the boundary condition (38) and the soil radial stress expression (39), the elastic solution for the axial force and bending moment of the liner under the air pressure are expressed as follows:

T

1 (C  F )[ Eu0 (5  3sin   3cos 2  )  4(1  v) pa r0 ] 4 (1  v)(C  F )  (1  v 2 )CF M 0

(45) (46)

21 4. Analytical solution for long-term effects 4.1. Long-term solution for ground displacements and liner internal forces without considering air pressure Similarly, the long-term analysis of the elastic deformation of the ground (no air pressure) can be simplified as a two-stage analysis: (1) a stage regardless of the hydraulic pressure; (2) a stage considering the action of the hydraulic pressure only. For the first stage, the solutions are exactly the same as those in Section 3.1. It should be noted that the Poisson's ratio of soil is taken as the actual value, for it no longer meets the limit value taken in the short-term analysis. Due to that, the ground displacements can be modified as:

1  E

 a0 3 1   2a2r  4(1  )b2r  r [c1r 2  c1 (1  ) ln r  9c3r 4  d1 (1  2 ) ln r  (15  12 ) d 3r 2 ]sin 

Ur 

[4a2r  8(1  )b2r ]sin  3

1

[12c3r 4 4(5  4 )d3r 2 ]sin 3 





1  [c1r 2  c1  c1 (1  ) ln r  9c3r 4 E d  (1  2 )(ln r  1)  3(1  4 )d r 2 ]cos 

U  1

(47)

2

3

[4a2r 3  4(1  2 )b2r 1 ]sin  cos  [12c3r 4  4(1  4 )d 3r 2 ]cos 3 



(48)

The corresponding pending parameters are as follows: 2 2 1 2 b h 1  k  1   CFr0  E  C  F  u0  5  3sin   3cos   a0  r0 4  C  F 1    1  2  CF

(49a)

c1   b r02

(49b)

1   4 c1'   k    b r0 8 1  

(49c)

22

d1' 

(49d)

1  F  6 1    b h 1  k  4 r0 4 1   F  3(5  6 )

(49e)

1  2 1   F  3  b h 1  k  2 r 4 1   F  3(5  6 ) 0

(49f)

1 1   F  4  5  4   b 1  k  r06 12 1   F  8(7  8 )

(49g)

1   F  8  1  k r 4 1 0 b 8 1   F  8(7  8 )

(49h)

a2'  

b2'  c3' 

1 1  2  b r02 4 1 

d3'  

Regardless of the compressed air, the axial force and bending moment of the axial liner per unit meter at the excavation boundary are listed below:

1 [ Eu0 (2  3sin   3sin 2  )  2 b hr0 (1  k )(1   )](C  F ) 4 (C  F )(1  )  (1  2 )CF 3 (3  4 ) b h(1  k ) 9  12v  r0   b (1  k )r02 sin  2 (1  ) F  3(5  6 ) (1  ) F  8(7  8 ) 3(3  4 ) b h(1  k ) 12  16v  r0 sin 2    b (1  k )r02 sin 3  (1  ) F  3(5  6 ) (1  ) F  8(7  8 )

(50)

3 (3  4 ) b h(1  k ) 2 r0 2 (1  ) F  3(5  6 ) 9  12   b (1  k )r03 sin  (1  ) F  8(7  8 ) 3(3  4 ) b h(1  k ) 2 2  r0 sin  (1  ) F  3(5  6 ) 12  16   b (1  k )r03 sin 3  (1  ) F  8(7  8 )

(51)

T

M 

For the stage subjected to the compressed air alone, the mechanics model for the long-term ground-liner interaction based on the Airy stress function [59] necessarily introduces boundary conditions as follows: Ur

 r

r  r0

r  r0

 U rs

r  r0

  w (hw  r0 sin  )   rs0  0

(52a) (52b)

23

r

  rs

0

(52c)

u   w (hw  r0 sin  )

(52d)

r  r0

r  r0

where u is the pore pressure;  r is the radial effective stress of soil; and  rs0 is the liner radial stress. Based on the solutions gained in the first stage, the displacement expressions under the hydraulic pressure can be deduced with the assumptions made (    w ;

h  hw ; k  1 [40]), which are the same as Eqs. (47) and (48). Besides, the corresponding pending parameters are modified as:

a0 

1   CF  h r2  C  F   1   CF w w 0 c1   w r02 1 1  2v  w r04 8 1 v 1 1  2 d1'   w r02 4 1  c1' 

a2  b2  a3  b3  0

(53a) (53b) (53c) (53d) (53e)

Furthermore, under the action of hydraulic pressure alone, the axial force and bending moment of the liner per unit meter at the excavation surface are solved below:

T 

CF  w hw r0 C  F  (1  v)CF M 0

(54) (55)

Similarly, the full long-term ground displacement and liner internal force (under undrained boundary) can be gained by superposing the results derived in the two stages.

24 4.2. Long-term solution for ground displacements and liner internal forces with considering air pressure Corresponding to the short-term solution using the two-stage analysis (i.e., the excavation process and the liner installation process), the boundary conditions of the excavation interface are modified as follows (including proposed by Park [43]), for the long-term effects will be in line with the dissipation of the pore pressures:

r  r Ur

r  r0

 U rs

r  r0

r  r0

r  r0

  rs

  rs

(56a)

r  r0

r  r0

0

(56b)

1   u0 (5  3sin   3cos 2  )+(U ri +U rii ) 4

r  r0

(56c)

in which U ri is the ground radial displacement during the excavation, U rii is the ground radial displacement during the liner installation. The expressions are the same as those in Section 3.1. The long-term solutions for the ground displacement under the action of the compressed air can be expressed as follows:

1  E

 a0 3 1   2a2r  4(1  )b2r  r [c1r 2  c1 (1  ) ln r  9c3r 4  d1 (1  2 ) ln r  3(5  4 )d 3r 2 ]sin 

Ur 

[4a2r 3  8(1  )b2r 1 ]sin 2  [12c3r 4 4(5  4 )d3r 2 ]sin 3 

(57)





1  [c1r 2  c1  c1 (1  ) ln r  9c3r 4 E d  (1  2 )(ln r  1)  3(1  4 )d r 2 ]cos 

U  1

3

[4a2r  4(1  2 )b2r ]sin  cos  3

(58)

1

[12c3r 4  4(1  4 )d3r 2 ]cos3 



Based on the boundary condition (56), the corresponding pending parameters can

25 be determined as follows:

CF 1 a0    b h 1  k    w hw  2  C  F 1    1  2  CF  1    Eu0  5  3sin   3cos 2    1   pa   r02  4r0  

c1    b   w  r02

(59a)

(59b)

1 1  2 c1    b 1  k    b   w  r04 8 1  

(59c)

1 1  2  b   w  r02 4 1 

(59d)

d1  a2  

1 1   F  9  b h 1  k  r04 4 1   F  15

(59e)

b2' 

1 1   F +6  b h 1  k  r02 2 1   F  15

(59f)

1 1   F  4 17  4   b 1  k  r06 12 1   F  8(7  8 )

(59g)

1 1   F  72  b 1  k  r04 8 1   F  8(7  8 )

(59h)

c3' 

d3'  

The long-term solutions for the axial force and bending moment of the liner at the excavation boundary surface are: 1 [ Eu0 (5  3sin   3cos 2  )  4(1  v) pa r0 ](C  F ) 3 b h(1  k )  r0 4 (C  F )(1   )  (1  2 )CF (1  ) F  15 3  12v   b (1  k )r02 sin  (1  ) F  8(7  8 ) 6 b h(1  k )  r0 sin 2  (1  ) F  15 4  16v   b (1  k )r02 sin 3  (1  ) F  8(7  8 )

T

(60)

26

3 b h(1  k ) 2 r0 (1  ) F  15 3  12   b (1  k )r03 sin  (1  ) F  8(7  8 ) 6 b h(1  k ) 2 2  r0 sin  (1  ) F  15 4  16   b (1  k )r03 sin 3  (1  ) F  8(7  8 ) M 

(61)

5 Case verification and parameter analysis 5.1. Case comparisons for ground deformation For the gap parameter, Lee et al. [58] firstly suggested a gap parameter g for the ground displacements. Loganathan and Poulos [38] gave a detailed algorithm for the gap parameter g. Following the above parameter definitions and assuming u0 = g/2, this paper illustrates two different boundary conditions: Boundary B-1: Only considering the uniform radial contraction without considering the ovalization deformation (i.e., BC-1 proposed by Park [43]). Boundary B-2: Considering both the uniform radial contraction and the final ovalization deformation (i.e., BC-4 proposed by Park [43]). Seven case studies, which were previously investigated by various researchers, are selected to compare the analytical predictions with the above-mentioned boundary conditions. Table 1 summarizes the characteristic soil properties, geometry of the tunnels, the construction methods, etc. The following parameters are set: the soil Poisson's ratio, v=0.3 (long-term effect); the liner modulus, Es = 25000 MPa; and the liner Poisson's ratio, νs = 0.2. Additionally, the liner thickness t for the Bangkok Tunnel and Belfast Sewer Tunnel are set as 0.1m; and the liner thickness t for the Central Interceptor Tunnel and Regent Park Tunnel are denoted by 0.2 m.

27 In order to verify the applicability of theoretical solutions for short-term vertical and horizontal displacements under no air pressure, the short-term ground settlement curves of Bangkok Tunnel are drawn in Fig. 4, under the boundary conditions mentioned above. As indicated in Fig. 4, the negative values represent the ground settlements, whereas the positive represent the uplift value. Fig. 5 shows the curve of the short-term ground horizontal displacement at a vertical section of the tunnel. For the Bangkok Sewer, the vertical section for calculation is 4 m away from the center of the tunnel, where the negative value means the lateral shift is on the side closer to the tunnel. Furthermore, this paper comparatively analyzes the effects of the ovalization deformation on the ground displacement, and then validates the analysis with the measured data. It can be concluded from Fig. 4 and Fig. 5: (1) different deformation patterns all have significant influences on the ground settlements and horizontal displacements. The predicted settlement in boundary B-2 is generally larger than the boundary B-1, especially above the tunnel centerline. Since the oval-shaped convergence deformation was not taken into account, the lateral shift value of boundary B-1 is smaller near the surface but larger as the depth increases compared with the measured data. Moreover, the lateral shift value dramatically increases near the tunnel section, and the maximum value is over larger than the measured value; (2) the settlement curve obtained from boundary B-2 aligns well with the measured data, especially the maximum settlement is closer to the measured value. At the end of the excavation surface, the width of the settlement trough obtained by boundary B-2 gradually

28 converges, and it does not significantly narrow. The increasing rate of the lateral shift value gained by boundary B-2 decreases over the depth, and the inflection point of the lateral shift curve shows up. Furthermore, the lateral shift value converges quickly near the tunnel and agrees well with the measured data. In summary, the solutions of boundary B-2 are more in line with the trends of the vertical and horizontal deformation of the ground. Therefore, the solutions of B-2 can be trusted to provide a conservative prediction for the short-term effects of the surface settlements and horizontal displacements. Similarly, the theoretical solutions for the ground settlements under no compressed air condition are verified by examples. The long-term surface settlement curves are shown in Figs. 6 and 8, and the settlement curves at the vertical centerline of the tunnel are shown in Figs. 7 and 9, in which comparisons were made between the analytical results and measured data obtained two months after the construction. Figs. 6-9 compare the measured data and the predicted long-term surface settlements of the two boundary conditions. The predicted surface settlements are larger than the measured value at the far field of the excavation surface. However, the predicted ground settlements at the vertical central axis of the tunnel in B-2 boundary are more consistent with the measured data. In general, the boundary B-2, based on the oval-shaped deformation pattern, is more suitable for predicting the long-term ground settlements for shallow tunnels. To validate the reliability of the theoretical solutions for short-term vertical displacements under air pressure, the short-term settlement curves of each tunnel

29 under the two boundary conditions are drawn in Figs. 10-14. The graphs indicate that the trends of the vertical displacement curves are basically consistent with the measured data. The oval-shaped deformation pattern in boundary B-2 is also more similar to the measured data under the compressed air condition. In conclusion, the theoretical solution can be used as a preliminary prediction for the short-term ground settlement due to the compressed air excavation. In order to investigate the effects of air pressure and long- and short-term period on the surface displacement, comparisons are made in Figs. 15 and 16 among the ground settlement curves and lateral shift curves obtained by theoretical solutions of various factors in boundary B-2. The lateral shift curves are shown in Figs. 15 and 16, respectively. The vertical section of the lateral shift curves is 4 m from the center of the tunnel. The relevant parameters are set as follows: the tunnel radius, r0 = 4m; the depth, h = 20m; the soil Young’s modulus, E = 30 MPa; the liner modulus Es = 25000 MPa; the shield gap, g = 50 mm; the groundwater table, hw = 20 m; the soil bulk density, γb = 8 kN/m3; the soil Poisson's ratio, v=0.3 (long-term effect); the soil Poisson's ratio, v=0.5 (short-term effect); the liner Poisson's ratio, vs = 0.2; the liner thickness, t = 0.3 m; and the lateral pressure coefficient, k = 0.5. From Figs. 15 and 16, it can be noted that the calculated value considering air pressure is greater than the value without considering air pressure, i.e., the action of the compressed air results in an increase of the ground deformation, which is consistent with the following construction process by the compressed air: when no compressed air is applied, the liner is installed immediately after the excavation and

30 provides timely support to the soil load and groundwater pressure; when compressed air is applied, the liner installation takes place after excavation. During the excavation to the liner installation, the compressed air supports the confining pressure of the groundwater alone, and the ground requires to be borne by itself to make sure its stability, which produces more ground displacements. In addition, it can also be noted that the value of the long-term solution is larger than the short-term, which also proves that the dissipations of the void pressures will lead to further increase in the ground deformation as the construction progresses. 5.2. Parameter analyses for liner displacement The relative displacement differences between the liner and excavation surface are uniformly expressed as the boundary condition (6). With the absence of air pressure, the short-term radial displacement of the liner at the circumference can be obtained by the following expression (Park [43]):

U rs

r  r0

=U r

r  r0

1 + u0 (5  3sin   3cos 2  ) 4

(62)

where U r and U rs are the radial displacements of the ground and liner, respectively; r0 is the radius of the tunnel; u0 is the radius difference between the shield surface

and segment liner. According to the assumption made that no relative slippage exists between the ground and liner, the short-term tangential displacement of the liner at the circumference can be expressed as:

Us

r  r0

=U

r  r0

(63)

Both the radial and tangential displacements around the liner can be obtained

31 above. In order to study the influence of the relative parameters on the liner displacements, this study builds on the short-term solutions for liner displacements under no air pressure, adjusting the tunnel radius r0 and soil Young’s modulus E. As shown in Figs. 17 and 18, the short-term elastic solution for the radial and tangential displacements of the liner along the circumference are obtained, based on the oval-shaped deformation pattern of boundary B-2. The basic parameters are set as follows: the tunnel radius, r0 = 4 m; the depth, h = 20 m; the soil Young’s modulus, E = 30 MPa; the liner modulus, Es = 25000 MPa; the shield gap, g = 50 mm; the groundwater table, hw = 20 m; the soil unit weight, γ =18 kN/m3; the soil Poisson's ratio, v = 0.5 (short-term effect); the liner Poisson's ratio, vs = 0.2; the liner thickness, t = 0.3 m; and the lateral pressure coefficient, k = 0.5. Fig. 17 shows the variation of the radial displacements of the liner along the circumference after adjusting the two parameters. It can be gained from Fig. 17: (1) the radial displacement of the liner is strictly symmetrical by the axis 90o/270o (the central axis of the tunnel); the radial displacement value shows negative over the entire circumference, which reflects the liner deformation characteristic of radial compression; (2) the radial negative displacement reaches its maximum at the 90o crown and gradually decreases towards both sides with the change of angle. The minimum negative displacement is at the bottom of 270o arch, which reflects the deformation trend of liner ovaliztion and overall settlement during the excavation; (3) the smaller the radius and stiffer the ground, the smaller the radial displacement of the

32 liner, and the more uniform the distribution of the displacement value in the circumferential direction, that is, the less obvious the ovalization deformation trend. As the radius increases and soil Young’s modulus decreases, the radial displacement of the liner becomes larger and the ovalization deformation tendency turns more obvious. The variation of radial displacements of the liner are obtained in Fig. 18 after adjusting the two parameters at the excavation surface along the circumference. From Fig. 18, we can see that: (1) the value of tangential displacement is 0 at the 90o arch and 270o bottom, which is taken as the axis boundary. The positive value occurs in the left of the tunnel, and the negative value occurs in the right of the tunnel; (2) the maximum value of the tangential negative displacement occurs near axis 30o, and the maximum positive value occurs near axis 150o; (3) the tangential displacements of the liner with small radii and tunnels in stiff clay are generally smaller. Besides, an overall increase of the tangential displacement can be obtained by increasing the radius and reducing the soil elasticity modules, and the maximum positive and negative tangential displacement values all increase significantly. 5.3. Parameter analysis for liner internal forces During the excavation, the liner internal forces are remarkably affected by the primary parameters such as the material properties of the ground and liner, the geometric characteristics of the tunnel, etc. In order to study the effects of these parameters on the theoretical calculation, this paper builds on the long-term solution for the axial force of liner considering air pressure. A long-term elastic solution for the

33 axial force of liner under the oval-shaped deformation pattern of boundary B-2 is obtained by adjusting tunnel depth h, tunnel radius r0, soil Young’s modulus E, and gap parameter g, (Fig. 19). The soil Poisson's ratio v is 0.3 (long-term effect) in the cases studied, and the rest of the basic parameters can be found according to the calculations in Section 5.2. Fig. 19 shows the variation of the axial force of liner per unit meter along the circumference of the excavation face after adjusting the four parameters. According to Eq. (29), the tunnel radius r0 can be used to characterize the variation of the air pressure generated by the shield with other parameters unchanged. For example, pa=220kPa when r0=2m; pa =230kPa when r0=3m; pa =240kPa when r0=4m; pa =250kPa when r0=5m; pa =260kPa when r0=6m. From Fig. 19, the axial force of the liner is strictly symmetrical by the axis 90o/270o (the central axis of the tunnel), which is demonstrated as an inverted “8” (the top circle is objectively smaller than the bottom). Affected by the oval-shaped deformation pattern at the periphery of the tunnel opening, the compression zone of the axial force (positive value) is distributed around the arch waist of both sides. The maximum compression value occurs slightly below the arch waist; and the tension zone (negative value) is distributed around the crown arch and the bottom where the maximum tension value occurs at the crown of 90o, and the axial force at the bottom of 270o arch is clearly smaller than at 90o. In addition, it can also be found from Fig. 19 that the influence of tunnel depth h and tunnel radius r0 on the axial force of liner is similar. As the depth h and the radius r0 increase, the compression zone is expanded while the tension zone is decreased.

34 Similarly, as the soil Young’s modulus E and gap parameter g increases, the tension zone is expanded while the compression zone is decreased. Besides, the increasing rate of the tension zone is clearly faster, which indicates that the more obvious the distribution of the axial force (the crown arch is under tension status, and the arch waist is under compression status) shows in stiff soil and large gap, the more obvious the trend of ovalization is. At the same time, the variation of the radius also reflects the influence of the air pressure on the axial force distribution of the liner. As shown in Fig. 19b, as the air pressure goes up, the compression zone of the axial force is dramatically expanded because the action of the confining pressure increases the uniform pressure at the periphery of the tunnel opening and reduces the tension of the axial force at the crown arch and the bottom to some extent. Based on the long-term solution for the bending moment of the liner considering air pressure, the tunnel depth h, tunnel radius r0, lateral pressure coefficient k, and soil weight γ are adjusted to obtain the long-term elastic solution for the circumferential bending moment of the liner based on the B-2 oval-shaped deformation pattern. Fig. 20 shows the variation of the circumferential bending moment of liner per unit meter after adjustment of the four parameters. The distribution of bending moment of the liner is strictly symmetrical by the coordinate axis 90o/270o in the type of an “8” in which the top circle is highly uniform to the bottom. The positive value (compression for outer side of liner, tension for internal side of liner) of bending moment occurs near the bottom and crown arch with the maximum value at the bottom of 270o. The negative value (tension for outer side of liner, compression for

35 internal side of liner) of bending moment occurs near the arch waist of both sides with the maximum values of 0o and 180o. Adjustments of tunnel depth h, tunnel radius r0, and soil unit weight γ has the same effect on the bending moment of the liner. Both the positive and negative bending moments are reduced as these parameters decrease, and the distribution of “8” is also oriented towards “0”, which is opposite to the influencing variation of the lateral pressure coefficient k. The smaller the lateral pressure coefficient, the more obvious the distribution pattern of “8”, and the positive and negative bending moment values are gradually increased. Similarly, the positive and negative bending moment values of the liner increases as the air pressure rises (Fig. 20b). 6. Conclusions This paper presents a closed-form method to predict the ground movements for the long- and short-term effects caused by shallow tunnels in saturated soils with liner considering the oval-shaped convergence deformation pattern and shield air pressure. On the basis of the above, this paper focuses on the liner stress-internal force relationship and establishes the elastic analytical solution for the liner internal forces of the oval-shaped deformation pattern. The meaningful work is to present a simplified and effective approach to analyze the displacement boundary condition of tunnel opening based on the oval-shaped convergence pattern and the interaction mechanics between the ground and segment liner. Furthermore, the above-mentioned aim is accomplished considering the shield air pressure and saturated soils. The proposed method is verified through comparisons with the published solutions by the

36 observed data. It has been demonstrated that the proposed method based on the oval-shaped convergence pattern can give a reliable prediction for the response of ground movements for shallow tunnels with or without the shield air pressure. The results indicate that the deformation pattern at excavation opening has a remarkable influence on the ground deformation, and the uniform convergence pattern may not predict correctly the ground deformation behavior. The method can be used to estimate the ground movements and liner internal forces in the preliminary design of shallow tunnels in different construction conditions and environments, including the shield excavation with air pressure and encountered saturated soils. The deformation pattern at the excavation opening has a significant impact on the ground displacement field. The oval-shaped deformation pattern increases the long- and short-term ground settlement values above tunnels, which aligns well with the measured data. Owing to the oval-shaped deformation, the lateral shift value increases slower gradually as the depth increases, and obtain a clear convergence trend near the tunnel. The oval-shaped deformation pattern at the excavation opening meets both the shield driven tunneling with and without air pressure. The solution for the ground deformation proposed in this paper can conservatively predict the longand short-term surface displacement values. The ground deformation that considers the air pressure is larger than the one that does not. The long-term ground deformation is larger than the short-term. As the construction process advances, the dissipation of pore pressure leads to further increase of the ground deformation. The analytical solutions for liner radial displacement and internal force are also based on the

37 oval-shaped convergence deformation pattern. The calculation curves for the liner radial displacement, axial forces and bending moments are strictly symmetrical according to the axis of 90o/270o (the central axis of the tunnel). The distribution of the liner radial displacement reflects the obvious ovalization and overall subsidence trend of the liner. The tunnel radius and soil Young’s modulus significantly affect the radial displacement of the liner. According to different lateral pressure coefficient of soil, the type of axial force is an inverted “8” along a circumference in which the top circle is obviously larger than the bottom. However, the type of bending moment is an “8” in which the top circle is generally uniform to the bottom. The values of lateral earth pressure coefficient significantly affect the axial forces and bending moments of the liners. In fact, it is not difficult to extend this proposed method to analyze other related problems, such as tunneling on existing the pile foundations, and adjacent the pipelines. It should be noted that the major limitation of the proposed method stems from the simplified assumptions of linearity and elasticity. Physically, the ground is perfectly elastic but generally non-linear and tunnel liner is not homogeneous, which cannot be considered by this analytical method. Besides, no attempts have been made in this study to investigate the relative sliding exists between the liner and ground. Though the analytical solution presented in this study is limited in elastic scope, it appears to be useful for the preliminary design of shallow tunnels in different construction processes and conditions. Advanced mechanisms such as the relative sliding failure and friction between the liner and ground, and advanced elasto-plastic

38 or elasto-viscoplastic constitutive models for ground, should be introduced into this study. Furthermore, with the help of numerical software and measure technology, relevant studies remain to be done to evaluate the ground deformation behavior and liner deformation status affected by tunneling. Therefore, further research on this subject is still required to more effectively evaluate the problem for ground-liner interaction mechanics. Acknowledgements： The authors acknowledge the pioneered achievements proposed by Prof. Antonio Bobet (Purdue University) and Engineer Wei-I. Chou (Professional Service Industries), their research is the important foundation of the theoretical solution in this paper. The authors acknowledge contributions to this revision paper proposed by Assistant Prof. Zhiwei Wang (State Key Laboratory for Track Technology of High-speed Railway, China Academy of Railway Sciences). The authors acknowledge the financial support provided by the National Natural Science Foundation of China for Key Program and General Program (No. 51738010, 41772331 and 41977247), and the Project Program of Key Laboratory of Urban Underground Engineering of Ministry of Education (No. TUE2017-04), and the Project Program of State Key Laboratory for Track Technology of High-speed Railway under research grant (No. 2018YJ181).

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47

Figure Captions

T*

Segment liner Gap (g)＞2T*

T* Tunnel section (A) Uniform radial [35,36]

(B) Oval-shaped [38]

Figure 1. Deformation patterns around tunnel section [35,36,38]: (a) Uniform radial [35,36]; (b) Oval-shaped [38]

48

2u0

u0

u0

u0

u0

u0

u0

(a) BC-1

(b) BC-2

2u0

2u0

u0 2

u0 2

u0 2

u0 2

u0 2 (c) BC-3

(d) BC-4

Figure 2. Boundary conditions around tunnel section [43,45]: (a) BC-1; (b) BC-2; (c) BC-3; (d) BC-4

49

Surface

Ground water h

hw

y

Ur

Uθ

σθ

σr

Ground (E, ν)

τrθ

σr

σθ

r θ r0

t Liner (Es, νs)

Figure 3. Analysis model of shallow tunnel with liner

x

50

0

0

Distance from tunnel centerline (m) 5 10 15 20

25

Settlement (mm)

-5

-10 Short-term B-1 (no air pressure) Short-term B-2 (no air pressure) Observed

-15

Figure 4. Short-term effect of boundary conditions (Bangkok Sewer Tunnel: surface settlement)

51

0

0

-2

4

Depth (m)

8

Horizontal displacement (mm) -4 -6 -8 -10 -12

-14

-16

Short-term B-1 (no air pressure) Short-term B-2 (no air pressure) Observed

12 16 20 24 28

Figure 5. Short-term effect of boundary conditions (Bangkok Sewer Tunnel: lateral displacement)

52

0

0

Distance from tunnel centerline (m) 5 10

15

Settlement (mm)

-2

-4

-6 Long-term B-1 (no air pressure) Long-term B-2 (no air pressure) Observed (two months after construction)

-8

-10

Figure 6. Long-term effect of boundary conditions (Regent Park Southbound Tunnel: surface settlement)

0

0

-10

Settlement (mm) -20

-30

-40

5

Depth (m)

10 15

Long-term B-1 (no air pressure) Long-term B-2 (no air pressure) Observed (two months after construction)

20 25 30 35

Figure 7. Long-term effect of boundary conditions (Regent Park Southbound Tunnel: ground settlement above tunnel centerline)

53

0

0

Distance from tunnel centerline (m) 3 6 9

12

-2

Settlement (mm)

-4 -6 -8 Long-term B-1 (no air pressure) Long-term B-2 (no air pressure) Observed (two months after construction)

-10 -12 -14

Figure 8. Long-term effect of boundary conditions (Regent Park Northbound Tunnel: surface settlement)

-5 0

Depth (m)

4

-10

Settlement (mm) -15 -20

-25

-30

Long-term B-1 (no air pressure) Long-term B-2 (no air pressure) Observed (two months after construction)

8

12

16

20

Figure 9. Long-term effect of boundary conditions (Regent Park Northbound Tunnel: ground settlement above tunnel centerline)

54

0

0

Distance from tunnel centerline (m) 1 2 3 4

5

Settlement (mm)

-5

-10

Short-term B-1 (air pressure) Short-term B-2 (air pressure) Observed

-15

-20

0

0

Distance from tunnel centerline (m) 1 2 3 4

5

Settlement (mm)

-5

-10

-15

Short-term B-1 (air pressure) Short-term B-2 (air pressure) Observed

-20

Figure 10. Short-term effect of boundary conditions (Belfast Sewer Tunnel-Array A: surface settlement)

55

0

0

Distance from tunnel centerline (m) 1 2 3 4

5

Settlement (mm)

-5

-10

Short-term B-1 (air pressure) Short-term B-2 (air pressure) Observed

-15

-20

Figure 11. Short-term effect of boundary conditions (Belfast Sewer Tunnel-Array B: surface settlement)

0

0

Distance from tunnel centerline (m) 10 20

30

Settlement (mm)

-30

-60

-90

-120

Short-term B-1 (air pressure) Short-term B-2 (air pressure) Observed

-150

Figure 12. Short-term effect of boundary conditions (Central Interceptor Tunnel-Section 6: surface settlement)

56

0

0

Distance from tunnel centerline (m) 5 10 15

20

Settlement (mm)

-40

-80

-120

Short-term B-1 (air pressure) Short-term B-2 (air pressure) Observed

-160

Figure 13. Short-term effect of boundary conditions (Central Interceptor Tunnel-Section 5: ground settlement at a depth of 6 m below surface)

Settlement (mm)

0

0

Distance from tunnel centerline (m) 4 8 12

16

-60

-120 Short-term B-1 (air pressure) Short-term B-2 (air pressure) Observed

-180

Figure 14. Short-term effect of boundary conditions (Central Interceptor Tunnel-Section 5: ground settlement at a depth of 12 m below surface)

57

0

Distance from tunnel centerline (m) 5 10 15

0

20

-5

Settlement (mm)

-10 -15 -20 -25 Short-term (no air pressure) Long-term (no air pressure) Short-term (air pressure) Long-term (air pressure)

-30 -35 -40

Figure 15. Surface settlement comparisons of base case with different conditions

0 5

Depth (m)

10

0

-5

Horizontal displacement (mm) -10 -15 -20 -25 -30

-35

-40

Short-term (no air pressure) Long-term (no air pressure) Short-term (air pressure) Long-term (air pressure)

15 20 25 30 35

Figure 16. Lateral displacement comparisons of base case with different conditions

58

90 10 120

60 -10

150

30

-30 -50

180

0

-70

210

330

240

300 270 (a) Tunnel radius

r0=2m r=2m r0=3m r=3m r0=4m r=4m r0=5m r=5m r0=6m r=6m

59

90 120

-5

60

-15 150

30

-25 -35

180

0

-45

210

330 Eu=20MPa Eu=25MPa

240

300 270

Eu=30MPa

Eu=35MPa Eu=40MPa E=20MPa E=25MPa E=30MPa E=35MPa E=40MPa

(b) Soil Young’s modulus

Figure 17. Influences of tunnel radius and soil Young’s modulus on radial liner displacement of five cases: (a) Tunnel radius; (b) Soil Young’s modulus

60

90 80 120

60 40

150

30

0 -40

180

0

-80

210

330

240

300 270 (a) Tunnel radius

rr=2m 0=2m rr=3m 0=3m rr=4m 0=4m rr=5m 0=5m rr=6m 0=6m

61

90 120

40

60

15 150

30

-10 -35

180

0

-60

210

330

240

300 270

Eu=20MPa E=20MPa Eu=25MPa E=25MPa Eu=30MPa E=30MPa Eu=35MPa E=35MPa Eu=40MPa E=40MPa

(b) Soil Young’s modulus

Figure 18. Influences of tunnel radius and soil Young’s modulus on tangential liner displacement of five cases: (a) Tunnel radius; (b) Soil Young’s modulus

62

90 2200

120

60 1400

150

30

600 -200

180

0

-1000

210

330

240

300 270

h=8m h=8m h=12m h=12m h=16m h=16m h=20m h=20m h=24m h=24m

(a) Tunnel depth

90 120

60

2000 1000

150

30

0 -1000

180

0

-2000

210

330

240

300 270 (b) Tunnel radius

rr=2m 0=2m rr=3m 0=3m rr=4m 0=4m rr=5m 0=5m rr=6m 0=6m

63

90 120

2500

60

1500

150

30

500 -500

180

0

-1500

210

330 240

300 270 (c) Soil Young’s modulus

Eu=20MPa E=20MPa Eu=25MPa E=25MPa Eu=30MPa E=30MPa E=35MPa Eu=35MPa E=40MPa Eu=40MPa

64

90 120

2800

60

1800

150

30

800

-200

180

0

-1200

210

330 240

300 270

g=30mm g=30mm g=40mm g=40mm g=50mm g=50mm g=60mm g=60mm g=70mm g=70mm

(d) Gap

Figure 19. Influences of tunnel depth, tunnel radius, soil Young’s modulus and gap on axial force of liner per unit meter of five cases: (a) Tunnel depth; (b) Tunnel radius; (c) Soil Young’s modulus; (d) Gap

65

90 120

400

60

200 150

30

0 -200

180

0

-400

210

330 240

h=8m h=8m h=12m h=12m h=16m h=16m h=20m h=20m h=24m h=24m

300 270 (a) Tunnel depth

90 800 120

60 400

150

30

0 -400

180

0

-800

210

330

240

300 270 (b) Tunnel radius

rr=2m 0=2m rr=3m 0=3m rr=4m 0=4m rr=5m 0=5m rr=6m 0=6m

66

90 600 120

60 300

150

30

0 -300

180

0

-600

210

330

240

300 270

(c) Coefficient of lateral soil pressure

k=0.1 k=0.1 k=0.3 k=0.3 k=0.5 k=0.5 k=0.7 k=0.7 k=0.9 k=0.9

67

90 500

120

60 250

150

30

0 -250

180

0

-500

210

330

240

300 270

3 γ=16kN/m γ=16kN/m^3 3 γ=18kN/m γ=18kN/m^3 3 γ=20kN/m γ=20kN/m^3 3 γ=22kN/m γ=22kN/m^3 3 γ=24kN/m γ=24kN/m^3

(d) Soil unit weight

Figure 20. Influences of tunnel depth, tunnel radius, coefficient of lateral soil pressure and soil unit weight on bending moment of liner per unit meter of five cases: (a) Tunnel depth; (b) Tunnel radius; (c) Coefficient of lateral soil pressure; (d) Soil unit weight

68 Table 1 Geometry, soil and other parameters of tunnels investigated [38,41]

Soil

Tunnel

Dept

Radi

modul

h

us

us

h/m

r0 / m

E /MPa

Bangkok

Regent Park Southbound

Regent Park Northbound

Belfast Sewer Array A

Belfast Sewer Array B

18.5

34.1

20.1

4.85

4.4

1.33

2.07

2.07

1.37

1.37

20

56

56

4

4

Soil unit weight γ /(kN/m 3

)

17

19

19

15

15

Wate

Ga

r

p

table

g

hw

/m

/m

m

17

81

29.8

15.8

3.4

3.2

23

17

23

30

Lateral pressure coefficie

23.5

3.14

5

15

20

100

1

1.5

1.5

0.7

0.7

0.7

Section-5 Central Interceptor Section-6

27

3.14

5

15

25

80

type

Constructi on method

nt k

Central Interceptor

Soil

0.7

Soft to stiff clay

London clay

London clay

Soft silty clay

Soft silty clay

Soft clay with silt

Soft clay with silt

Excavated without air pressure Excavated without air pressure Excavated without air pressure Excavated with air pressure Excavated with air pressure Excavated with air pressure Excavated with air pressure