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Analytical solutions for nonlinear consolidation of soft soil around a shield tunnel with idealized sealing linings
Computers and Geotechnics 61 (2014) 144–152
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Analytical solutions for nonlinear consolidation of soft soil around a shield tunnel with idealized sealing linings Yi Cao, Jun Jiang ⇑, Kang-He Xie, Wei-Ming Huang College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China
a r t i c l e
i n f o
Article history: Received 23 February 2014 Received in revised form 21 May 2014 Accepted 22 May 2014
Keywords: Shield tunnel Nonlinear consolidation Analytical solution Sealing linings
a b s t r a c t This paper presents the analytical solutions for nonlinear consolidation of soft soil around a shield tunnel with idealized sealing linings. By introducing the empirical relation between permeability and compressibility, along with the conformal transformation, the governing equations of nonlinear consolidation are established, and the corresponding analytical solutions are derived. Then, the Terzaghi consolidation solutions are derived from the degenerate governing equation of nonlinear consolidation. Through the predictions of different consolidation theories in both completely permeable and impermeable lining conditions, the influences of a tunnel acting as a drain and impacting the dissipation of pore pressure, degree of consolidation, long-term ground settlements and ground settlement rates are investigated. During the early stages of consolidation, the case studies reveal that the predictions made by this study strongly agree with the field data when a completely permeable lining is applied. This study confirms that a tunnel acting as a drain can accelerate the consolidation of soil and enlarge soil deformation due to consolidation. During long term consolidation, a notable nonlinearity of the soil consolidation is exhibited by a small and gradually decreasing settlement rate, showing agreement with the tendency of field data from the impermeable conditions. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Many theoretical and field studies have been performed on tunneling induced ground settlement, yet most of these studies are concerned with short-term ground movement induced by tunnel construction [1]. However, the ground settlement around tunnels in soft clays continuously increases for many years after construction, especially around those built in coastal areas. Peck proposed an empirical formula for tunneling settlement [2], and then, Fang et al. proposed a time dependent equation for ground settlement above a tunnel’s center line [3]. Because of the large difference between the hydraulic head inside and outside tunnel linings after the construction of a tunnel, as well as joint leaking caused by segmented construction and differential settlement, Wongsaroj et al. concluded that tunnels in soft clays actually act as drains [4], which introduces a new drainage boundary condition that leads to long-term reductions in pore pressure and associated consolidation settlements. Mair and Taylor argued that the resulting settlement profile at the ground surface will tend to be considerably wider than the profile associated with construction [5]. ⇑ Corresponding author. Tel.: +8613957115706. E-mail addresses: [email protected] (Y. Cao), [email protected] (J. Jiang), [email protected] (K.-H. Xie), [email protected] (W.-M. Huang). http://dx.doi.org/10.1016/j.compgeo.2014.05.014 0266-352X/Ó 2014 Elsevier Ltd. All rights reserved.
Further research has confirmed, through measurements of pore pressures around tunnels, that tunnels constructed in soils with low-permeability act as drainage boundaries [6,7]. Moreover, various case studies and numerical analyses reveal that segmentally lined tunnels still act as drains when grouted [8,9]. Since then, many studies on this problem have been performed by using analytical and numerical tools and by considering the lining as a homogeneous permeable body [10–16]. However, the nonlinear behavior of soil in long-term consolidation has not been taken into consideration by the aforementioned studies. Based on an enormous amount of field measurements and theoretical research work, Shirlaw concluded that the proportion of long-term consolidation induced by construction disturbance to total settlement is between 30% and 90% [17]. Considering the immediately occurring construction settlement, the proportion of consolidation to total settlement after tunnel operation is even higher. Duncan found that the permeability and compressibility of soft clays change very much during long-term consolidation, causing a serious deviation from using the conventional consolidation theory in predicting the long-term settlement of soft clays [18]. As an effective tool to overcome the limitations of the conventional consolidation theory, the nonlinear consolidation theory can well predict the nonlinear behavior of soft clays [19–22].
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This paper idealizes the linings of tunnels as being completely permeable or completely impermeable, which represents the upper and lower limits of the induced soil consolidation behavior, and investigates how a tunnel acting as a drain affects the long-term behavior of the surrounding soil. In this paper the tunnel support is assumed to be rigid, so that the ground deformation around the tunnel section is short-term behavior compared with the effects of nonlinear consolidation. It is assumed that most of the ground deformation around the tunnel section is already completed during tunneling. Thus the tunnel support is not taken into account in the analysis presented. By introducing the classic e-lg k relation of nonlinear consolidation, and the conformal transformation, the corresponding consolidation equation is established. Then, the analytical solutions are developed for a long-term nonlinear consolidation with idealized drainage boundaries. After that, the obtained nonlinear solutions for excess pore pressure, degree of consolidation, and ground settlements are discussed by comparing them with the conventional Terzaghi consolidation solution corresponding to these special cases. Through comparison with settlement curves of field measured data and a modified Peck formula, the rationality of the upper and lower limits by the presented analytical solutions of both idealized lining conditions is discussed. 2. Statement of the problem The geometry of a circular tunnel constructed in soft clay with an idealized lining is shown in Fig. 1. The radius of the tunnel buried at a depth ‘h’ in soft clay is r, and the origin of the coordinate is at the ground surface above the tunnel. Considering Davis’s hypothesis for nonlinear consolidation of soil [19], the basic assumptions made in developing the solutions presented in this study are stated below: 1. The longitudinal length of the tunnel is large enough to meet plane strain conditions. 2. The soil around the tunnel is regarded as a half space with a cavity. 3. The soil around the tunnel is an isotropic saturated porous medium, and the permeability and compressibility coefficients are subjected to the e-lg k and e-lg r relation, respectively, during the consolidation. 4. Soil particles and pore water are incompressible. Darcy’s law is valid. The excess pore pressure at the ground surface and at an infinite depth is zero. 5. Soil deformation during consolidation is small, neglecting the effect on the coordinate system.
6. Soil deforms freely, neglecting the arch effect of soil itself and the tunnel. 7. The construction of a shield tunnel only causes excess pore pressure, and does not change the distribution of total vertical stress. 3. Solutions for nonlinear consolidation of soft clay around tunnel 3.1. Governing equation Based on basic assumption (3), the permeability and compressibility coefficients of the soil are respectively subjected to an empirical nonlinear relation by Davis, Bardon, Mesri, and Xie [19–22], as below:
0 r e ¼ e0 C c lg 0 r0 ks e ¼ e0 þ C k lg ks0
ð1Þ ð2Þ
where e is the void ratio, e0 is the initial void ratio, r00 is the initial effective stress, r0 is the effective stress, ks0 is the initial permeability coefficient of the soil, ks is the permeability coefficient of the soil, Cc is the initial compression index, and Ck is the initial permeability index. Then, from Eqs. (1) and (2), it can be obtained that
ks ¼ ks0
r00 =r0
CCc
k
1 @e r0 ¼ mm0 00 mm ¼ 1 þ e0 @ r0 r
ð3Þ ð4Þ
where mv is the coefficient of compressibility, and mm0 ¼ C c =ð1 þ e0 Þr00 ln 10 is the initial coefficient of compressibility. Then, substituting mv0 into Eq. (4), we can obtain
mm ¼
Cc 1 ð1 þ e0 Þ ln 10 r0
ð5Þ
From the free strain assumption, the continuity equation of saturated soil can be derived
1 @ @ Du 1 @ @ Du @ em þ ¼ ks ks @x @y cw @x cw @y @t @ em 1 @e ¼ 1 þ e0 @t @t
ð6Þ ð7Þ
where cw is the bulk density of water, Du is the excess pore pressure, and ev is the body strain. From the principle of effective stress, we can obtain
@ r0 @ Du ¼ @t @t
ð8Þ
Then, substituting Eqs. (5) and (8) into Eq. (7), we can obtain
@ em Cc 1 @ Du ¼ @t ð1 þ e0 Þ ln 10 r0 @t
ð9Þ
Assuming that the compressibility and permeability are decreasing all along with increasing pressure, (as done by Davis) [19], i.e., Cc/Ck = 1, by substituting Eqs. (3) and (9) into Eq. (6), the governing equation is then derived as
0 0 1 @ r @ Du 1 @ r @ Du þ ks0 00 ks0 00 cw @x r @x cw @y r @y ¼ Fig. 1. Geometry of a tunnel in soft clay.
Cc 1 @ Du ð1 þ e0 Þ ln 10 r0 @t
Using the substitution w:
ð10Þ
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w ¼ ln
r0
Du0 þ r00
ð11Þ
ffi Duc jpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þðHyÞ2 PH
where u0 is the initial excess pore pressure. Then, the governing equation can be simplified as:
! @ w @ w @w þ 2 ¼ @x2 @y @t 2
Cm
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h rs 2ðq0 cos hÞ 1þq2 2q cos h ; i:e:; Duc ¼ q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hr 2ðq0 cos hÞ 1þR2 2R cos h ¼0 Duc jqPq0 ¼ 0
ffi ¼q Duc jr6pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þðHyÞ2 6H
ð16Þ
2
ð12Þ
where Cm = ks0/(cwmm0) is the coefficient of consolidation.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where rs is x2 þ ðh yÞ , and q0 is the boundary of the distribution of excess pore pressure in the complex plane. The drainage boundary condition of the interface between the tunnel and the surrounding soil can be idealized in two forms due to different lining conditions:
3.2. Complex conformal transformation Because of the difficulty in analyzing the nonlinear consolidation of soft clay surrounding a tunnel in rectangular coordinate system, the considered region in the physical plane Z = x + yi is mapped to a circular ring region in the image plane f by using the complex conformal transformation, where the outer boundary with radius 1 of the ring region is the ground surface in the Z plane and the inner boundary with radius R of the ring region is the interface between the tunnel and soil, as shown in Fig. 2. The conformal mapping function by Verruijt [23] is given as below:
z ¼ xðfÞ ¼ ih
1 R2 1 þ f 1 þ R2 1 f
ð13Þ
where h is the tunnel depth, and R is the radius of the tunnel in image plane f, in which
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r R ¼ h h r2
ð14Þ
Then, the governing equation after conformal mapping can be obtained:
Cm
! @2w 4a2 @w þ 2 ¼ 2 2 @ @t g 2 @n 2 ð1 nÞ þ g
@2w
where a ¼
ð15Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h r2 .
3.3. Boundary and initial conditions According to basic assumption (4), the excess pore pressure at the ground surface and at an infinite depth is zero. The magnitude of the initial excess pore pressure at the interface between the tunnel and soil is known as q, and the distribution of excess pore pressure by the tunnel construction is assumed to be approximately uniform around the tunnel hoop and linear along the radial direction. Then, the initial distribution of the excess pore pressure from tunnel construction Duc, can be described as follows:
(1) Completely permeable, i.e., pore pressure is zero at the interface between the lining and the soil. (2) Completely impermeable. When completely permeable lining is applied, the shield tunnel introduces a new drainage boundary for soil consolidation, which strongly affects the dissipation of pore pressure. When the final steady flow state is reached, the decrease of hydrostatic pore pressure in the surrounding soil, along with the dissipation of excess pore pressure by the tunnel construction, defines the incremental effective stress, which is stated as follows:
Du0 ¼ Duc þ u0 us ¼ q
h rs þ cw y us hr
ð17Þ
where u0 is the hydrostatic pore pressure before the tunnel construction, us is the pore pressure when a steady flowing state is reached, and rs is the distance of calculation point from tunnel center. When tunnel lining is assumed to be completely permeable, the pore pressure at the soil-tunnel lining interface is zero, and the total water head is just the position hydraulic water head. According to the analytical solution of lining flow and soil flow by Laver [24]:
Qt ¼
2pr s ks us ujq¼R
ð18Þ
cw ðrs r2 Þ
where Qt is the lining flow per unit tunnel length per unit time, and r2 is the outer radius of the tunnel. Under the condition of completely permeable lining, u|q=R, the pore pressure at the soil-tunnel interface is zero. Soil flow per unit tunnel length per unit time along the flowing interface of which the distance from the tunnel lining is rs r2, can be given as
Qs ¼
2pks ½hcw us s cw ln 1 þ hr rs
ð19Þ
Given that the water flow through different seepage interfaces in the soil going into the tunnel are equal, the distribution of pore pressures of a steady flow state can be derived from Eqs. (18) and (19) as follows:
ðr s r 2 Þ us ¼ hcw sÞ r s ln 1 þ ðhr þ ðr s r 2 Þ rs
ð20Þ
Then, the 1st idealized boundary (completely permeable) with the initial conditions can be respectively described in the Z plane and f plane as
Dujt¼0 ¼ Du0 Dujq¼1 ¼ 0 ; i:e:; Dujq¼R ¼ 0 Dujt¼1 ¼ 0 Fig. 2. The region after conformal mapping.
wðq; h; 0Þ ¼ ln
r00
Du0 þr00
wð1; h; tÞ ¼ 0
0
wjq¼R ¼ ln Du0rþr0
0
wjt¼1 ¼ 0
q¼R
¼0
ð21Þ
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Then, the 2nd idealized boundary (completely impermeable) with the initial conditions can be respectively described in the Z plane and f plane as
Dujt¼0 ¼ Du0 Dujq¼1 ¼ 0
; i:e: @ Du
¼0 @q
wðq; h; 0Þ ¼ ln
Dujt¼1 ¼ 0
wjt¼1 ¼ 0
q¼R
Du0 þr00
wð1; h; tÞ ¼ 0
@w
@ Du
¼ @w @q
@u @ q
q¼R
r00
Then, substituting the characteristic value kn into Eq. (22), the following can be derived:
@T n C m k2n þ Tn ¼ 0 @t 4a2
ð30Þ
By using the Laplace transform, we can get
q¼R
¼0
ð22Þ Tn ¼ e
C m k2 nt 4a2
ð31Þ
Then, we can obtain
wðq; h; tÞ ¼ Wðq; hÞTðtÞ ¼
3.4. The derivation of the general solution
1 X W nT n
ð32Þ
0
From the governing equation and boundary conditions stated above, using the method of separation of variables, i.e., w(q, h, t) = W(q, h)T(t), the governing equation can be transformed to:
( @ 2 W 1 @W 1 @ 2 W k2 @T C m k2 þ þ 2 þ W ¼0 þ 2 T ¼0 2 2 2 @q @t 4a q @ q q @h ð1 þ q2 2q cos hÞ ð23Þ where k is the characteristic value, and using the substitution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ 1= 1 þ q2 2q cos h, we can obtain:
From Eq. (11), the excess pore pressure can be obtained:
Duðq; h; tÞ ¼ ðDu0 þ r00 Þð1 ew Þ
ð33Þ
3.6. The analytical solution for the 2nd idealized boundary condition After separation of variables, the 2nd idealized boundary condition is
Wjq¼1 ¼ 0
¼0
ð34Þ
@W
@q
q¼R
@ 2 W 1 @W þ þ k2 W ¼ 0 @ v2 v @ v
ð24Þ
The general solution of Eq. (24) is
WðvÞ ¼ AJ 0 ðkvÞ þ BN0 ðkvÞ
ð25Þ
where J0(kx) is a zero-order Bessel function, N0(kx) is a zero-order Neumann function, and A and B are undetermined coefficients of the boundary conditions.
Substituting Eq. (34) into general solution (25), the characteristic equation for the 2nd idealized boundary condition can be obtained:
k k
J pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffi N0 pffiffiffiffiffiffiffiffiffiffiffiffiffi
0 22 cos h 22 cos h
¼0
J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
N
1 1 2 2 1þR 2R cos h
W n ðq
After the separation of variables, the 1st idealized boundary condition is
ð26Þ
Substituting Eq. (26) into general solution (25), the following equations can be obtained
8 k k ffi þ BN 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼0 > < AJ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi 22 cos h 22 cos h > k k : AJ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ BN 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼0 2 2 1þR 2R cos h
3 kn ! J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 6 7 k k 1þR2 2Rcos h n n N 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ¼ Gn 6 4J0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ q2 2q cos h 1 þ q2 2q cos h 5 kn N pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
; hÞ ¼ Gn W 0n
Wjq¼R ¼ 0
1þR 2R cos h
Then, we can obtain the general solution as:
3.5. The analytical solution for the 1st idealized boundary condition
Wjq¼1 ¼ 0
ð35Þ
1
1þR2 2Rcos h
ð36Þ
where Gn is an undetermined coefficient defined by the boundary conditions. Similarly, we can obtain
Tn ¼ e
C m k2 nt 4a2
ð37Þ
Then, the excess pore pressure can be obtained
ð27Þ
Duðq; h; tÞ ¼ ðDu0 þ r00 Þð1 ew Þ
ð38Þ
1þR 2R cos h
Considering Eq. (26), k needs to satisfy the following characteristic equation to obtain nonzero solutions of A and B:
3.7. The solution for the undetermined coefficient Gn
k k
J pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffi N0 pffiffiffiffiffiffiffiffiffiffiffiffiffi
0 22 cos h 22 cos h
¼ 0
J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k ffi ffi
N0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 2 2
According to the nature of the Bessel function, the characteristic equations with different characteristic values are weighted orthogonally in the interval of [R, 1]:
1þR 2R cos h
ð28Þ
Z
1þR 2R cos h
The characteristic equation is denoted as f(k) = 0, which is an even function of k, ignoring these negative roots. After gradually searching its arranged roots sequence fk1 ; k2 ; k3 kn g from zero, the general solution can be obtained as 3 kn ! J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 0 6 7 k k 1þR2 2R cos h n n N0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ¼ Gn 6 4J0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ q2 2q cos h 1 þ q2 2q cosh 5 kn N pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
; hÞ ¼ Gn W 0n
W n ðq
0
1þR2 2R cosh
ð29Þ
where Gn is an undetermined coefficient determined by boundary conditions.
1
W n ðvÞW m ðvÞvdv ¼ 0 n – m
ð39Þ
R
By the initial condition
0 Du0 ¼ Duðq; h; 0Þ ¼ ðDu0 þ r00 Þ 1 eRGn W n T n ð0Þ 0 ¼ ðDu0 þ r00 Þ 1 eRGn W n ðvÞ
ð40Þ
We can obtain:
1 X r00 Gn W 0n ðvÞTð0Þ ¼ ln Du0 þ r00 1
ð41Þ
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where Gn is as stated below:
From the nature of the Bessel function, we can obtain
ln Du0 þr0 W 0n ðvÞvdv R 0 Gn ¼ R 1 02 W n ðvÞvdv R R 1 r00 0 qcos h ln Du0 þr0 W n ðvÞ 2 dq R 0 ð1þq2 2q cos hÞ ¼ R 1 02 qcos h W n ðvÞ 2 dq R ð1þq2 2q cos hÞ R1
r00
R1 Gn ¼
ð42Þ
3.8. The average degree of consolidation according to the excess pore pressure The average degree of consolidation according to the excess pore pressure, Up, is given by
R 2p R 1
rðDu0 Dun Þdrdh R R 2p R 1 r Du0 drdh R 0 R 2p R 1 rðDu0 þ r0 Þð1 ew Þdrdh ¼ 1 0 R R 2p R 1 0 r Du0 drdh R 0 0
Up ¼
ð43Þ
3.9. The solution for consolidation ground settlement The volumetric strain of soil is given by
em ¼ ðe0 eÞ=ð1 þ e0 Þ
ð44Þ
Substituting Eq. (1) into Eq. (44), we obtain
em ¼
0 0 Cc r Cc r þ Du0 Du lg 0 ¼ lg 0 1 þ e0 r0 1 þ e0 r00
ð45Þ
From the generalized Hooke’s Law and plane strain assumption, the vertical strain of soil at any particular time and point can be obtained as
K 0 þ ð1 mÞ ey ðx; y; tÞ ¼ em ð1 2mÞðK 0 þ 1Þ
ð46Þ
where K0 is the coefficient of earth pressure at rest, and t is Poisson’s ratio of the soil. Then, the surface settlement S(t) above the tunnel at any time t and distance from the tunnel center line x can be obtained:
SðtÞ ¼
Z
0
ey ðx; y; tÞdy
ð47Þ
ðhþrÞ
4. Degeneration of the solution If basic assumption (3) is changed following the Terzaghi consolidation theory, it can be stated as ‘‘the soil around the tunnel is an isotropic saturated porous medium, and the permeability and compressibility coefficients of the soil surrounding the tunnel are constant during consolidation.’’ Then, the nonlinear consolidation governing Eq. (10) is degenerate to Terzaghi’s consolidation governing equation as shown below:
cm
@ 2 Du @ 2 Du þ @x2 @y2
! ¼
@ Du @t
ð48Þ
where cm = k/(cwmm0). When the same boundary and initial conditions, as well as conformal mapping, are applied, the corresponding analytic solutions for two idealized sealing linings can be derived as:
Duðq; h; tÞ ¼
1 X Gn W n T n n¼1
ð49Þ
Du0 W n ðvÞvdv ¼ R1 2 W n ðvÞvdv R
R
R1
qcos h Du0 W n ðvÞ 2 dq ð1þq2 2q cos hÞ R1 2 qcos h W n ðvÞ 2 dq R 2
R
ð50Þ
ð1þq 2q cos hÞ
The corresponding analytic solutions for degree of consolidation and ground settlement can be derived from the former statement. 5. Prediction of long-term behavior of soil around a shield tunnel To investigate the rationality of the presented solutions, the predictions of the presented analytic solutions are compared with the corresponding solutions of Terzaghi’s consolidation theory and field data from Shanghai Metro Line No. 1 by Qu [25]. The length of Shanghai Metro Line No. 1 is 14.6 km, the diameter of the tunnel is 6.2 m, and the thickness of overlay soil ranges from 6 m to 8 m. The tunnel is located in saturated mucky clay, which is a typical soft clay. At the calculated cross section, the tunnel center is located at the depth of 11 m, around which the soft clay layers are equivalent to a homogeneous soil with an initial permeability coefficient k0 = 2.1 109 ms1, an initial compression index Cc = 0.25, an initial void ratio e0 = 1, a Poisson’s ratio of soil t = 0.36, and a coefficient of earth pressure at rest K0 = 0.43. The solutions of nonlinear consolidation of soft clay around the tunnel with idealized completely permeable and completely impermeable lining for excess pore pressure, degree of consolidation, settlement rates, etc. are compared with the corresponding solutions of Terzaghi’s consolidation theory. The predictions of ground settlement by nonlinear and Terzaghi consolidation theories are compared with the field data. The predictions are denoted as below: (A) Solutions of Terzaghi consolidation when the lining is idealized as completely permeable, PT. (B) Solutions of Terzaghi consolidation when the lining is idealized as completely impermeable, IPT. (C) Solutions of presented nonlinear consolidation when the lining is idealized as completely permeable, PN. (D) Solutions of presented nonlinear consolidation when the lining is idealized as completely impermeable, IPN. (E) Pore pressure when steady flow state is reached, when the tunnel lining is completely permeable, SFPP. (F) Hydrostatic Pore Pressure, HPP. (G) Field data of ground settlement from Shanghai Metro Line No. 1, FS. 5.1. Dissipation of excess pore pressure The predictions of excess pore pressure varying with depth above the center line of the tunnel by the presented nonlinear consolidation solutions and Terzaghi consolidation solutions with idealized linings are shown in Fig. 3(a–d). Fig. 3 illustrates that whether permeable or impermeable lining conditions are applied, the predicted dissipation of the excess pore pressure by nonlinear consolidation is slower than that of Terzaghi consolidation. Along with the dissipation of excess pore pressure and increase of effective stress, the changes in the soil permeability and compressibility in these two consolidation theories are very different. The sealing condition determines the reduction of pore pressure during consolidation. The reduction of pore pressure when lining is completely impermeable is just the excess pore pressure due to tunnel construction, i.e., Duc, while the reduction of pore pressure when lining is completely permeable is Duc + HPPSFPP.
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At the early stages of consolidation, if the completely impermeable lining condition is applied, the dissipations of pore pressure by both consolidation theories are relatively closer than those of the completely permeable lining condition. Then, the dissipation of pore pressure by Terzaghi consolidation is much quicker than that by nonlinear consolidation, which means that the deviation between Terzaghi and the presented solutions increases with time. This deviation can be explained by a decrease in the soil permeability and compressibility, which slow down the dissipation of pore pressure. However, in the Terzaghi consolidation, the soil permeability and compressibility is constant. On the contrary, at the early stage of consolidation (t = 0– 100 d), when the completely permeable lining condition is applied, the dissipations of pore pressure by Terzaghi consolidation is also quicker than nonlinear consolidation, but the deviation of pore pressure by these two consolidation theories decrease with time. At t = 400 d, the consolidation of soil around the tunnel is almost complete, the excess pore pressure from both consolidation theories has dissipated by approximately 90%. It can be concluded that
the tunnel acting as a drain accelerates the dissipation of pore pressure. The closer to the completion of consolidation, the smaller the deviation between the pore pressures predicted by both consolidation theories is. 5.2. Growth of the degree of consolidation The predictions of the average degree of consolidation with time by the presented nonlinear consolidation solutions and Terzaghi consolidation solutions with idealized linings are shown in Fig. 4. Then, it can be similarly concluded from Fig. 4, as from Fig. 3 (a–d), that at the early phases of consolidation of soil around the tunnel with an impermeable lining, the growth of the two consolidation theories are very close. In the analysis, with decreasing compressibility and permeability of soil, the difference between the corresponding average degree of consolidation by nonlinear consolidation and Terzaghi consolidation reaches its maximum at t = 400–600 d. When tunnel lining is completely permeable, and
Pore pressure, u (kPa) 50
100
200
PT--Pore pressure in Permeable lining by Terzaghi theory IPT--Pore pressure in imPermeable lining by Terzaghi theory PN--Pore pressure in Permeable lining by nonlinear theory IPN-- Pore pressure in imPermeable lining by nonlinear theory SFPP-- Pore pressure when steady flowing state HPP-- Hydrostatic Pore Pressure
5
depth, z (m)
Pore pressure, u (kPa)
150
10
tunnel
0 0
15
20
50
200
PT--Pore pressure in Permeable lining by Terzaghi theory IPT--Pore pressure in imPermeable lining by Terzaghi theory PN--Pore pressure in Permeable lining by nonlinear theory IPN-- Pore pressure in imPermeable lining by nonlinear theory SFPP-- Pore pressure when steady flowing state HPP-Hydrostatic Pore Pressure
10
tunnel
50d
(a)
20
(c)
300d
Pore pressure, u (kPa) 100
Pore pressure, u (kPa)
150
PT--Pore pressure in Permeable lining by Terzaghi theory IPT--Pore pressure in imPermeable lining by Terzaghi theory PN--Pore pressure in Permeable lining by nonlinear theory IPN-- Pore pressure in imPermeable lining by nonlinear theory SFPP-- Pore pressure when steady flowing state HPP-- Hydrostatic Pore Pressure
10
tunnel
0 0
200
50
100
10
tunnel
15
15
100d
(b)
20
150
200
PT--Pore pressure in Permeable lining by Terzaghi theory IPT--Pore pressure in imPermeable lining by Terzaghi theory PN--Pore pressure in Permeable lining by nonlinear theory IPN-- Pore pressure in imPermeable lining by nonlinear theory SFPP-- Pore pressure when steady flowing state HPP-- Hydrostatic Pore Pressure
5
depth, z (m)
50
5
depth, z (m)
150
15
0 0
20
100
5
depth, z (m)
0 0
(d)
Fig. 3. Distribution of pore pressure along depth.
600d
150
Y. Cao et al. / Computers and Geotechnics 61 (2014) 144–152
considering the whole time period of consolidation, the growth of the predicted average degree of consolidation by Terzaghi’s solution is faster than that by the presented nonlinear solution. The curves are very close when t = 400 d. This result illustrates that when a tunnel acts as a drain, it accelerates the consolidation of soil significantly. 5.3. Ground settlement and settlement rate at the tunnel center line due to soil consolidation with time The predicted ground settlement and settlement rate by presented nonlinear consolidation solutions and Terzaghi consolidation solutions with idealized linings, and field data of ground settlement from Shanghai Metro Line No. 1 are shown in Figs. 5 and 6, respectively. It is shown in Fig. 5 that the predicted ground settlement of the tunnel with a completely permeable lining is much larger than that of the tunnel with a completely impermeable lining, using either the presented nonlinear consolidation solution or the Terzaghi consolidation solution. According to Eqs. (45) and (47), the dissipation of pore pressure determines the soil deformation due to consolidation. Together with Eq. (17), it is concluded that tunnel acting as a drain enlarged the soil deformation due to consolidation. Moreover, this result illustrates that at the early stages of consolidation, the predicted ground settlements by both consolidation solutions in the permeable lining condition are very close to the field data from Shanghai Metro Line No. 1, revealing that at the early stages of soil consolidation, the completely permeable lining condition is closer to the real drainage condition. After approximately 100 d of consolidation, the settlement rate of field data decreased, with a tendency of ground settlement being closer to the predictions by the presented nonlinear solution and Terzaghi consolidation solution in the impermeable lining condition. At this phase, the field measured ground settlements were between the predictions of the two idealized sealing linings, with a curve closer to the permeable lining condition. Fig. 6 reveals that at the early stages of consolidation (approximately 0–150 d), the predicted settlements rates in completely permeable lining conditions by either of these consolidation theories are much larger than in impermeable lining conditions. Similar to the conclusion from Figs. 3 and 4, the tunnel acts as a drain, accelerating the consolidation of soil around the tunnel. During the soil consolidation in permeable lining conditions, there is a rapid decrease in the predicted settlement rate by both of these consolidation theories. In impermeable lining conditions, the curves of the settlement rates are smoother, indicating a much slower decrease, while the prediction by nonlinear consolidation
Fig. 6. Settlement rates due to soil consolidation around the tunnel with idealized linings.
0.8
has a smaller decrease than that given by Terzaghi consolidation. After consolidation of approximately 200 d, the predicted settlement rates by the nonlinear consolidation solution are larger than those of Terzaghi consolidation solution. According to the discussion above, when the same drainage boundary condition is applied during the first stage of consolidation (t < 100 d), the deviation between the predictions of ground settlement by nonlinear solution and the Terzaghi solution is not that significant. Then, the deviation increases after the first stage of consolidation. When consolidation is nearly finished, the deviation decreases. It is noted that the deviation for the permeable lining condition is much smaller than that in impermeable lining conditions.
0.6
5.4. Settlement curve with time
1.0
Average degree of consolidation, U
Fig. 5. Ground settlements due to soil consolidation around the tunnel with idealized linings.
The hyperbolic model for time dependent ground settlement above the tunnel center line proposed by Fang et al. [3] is shown as follows:
0.4 PT IPT PN IPN
0.2 0.0
0
200
400
600
800
time, t (day) Fig. 4. Growth of the degree of consolidation with time.
Smax ðtÞ ¼ 1000
t a þ bt
ð51Þ
where Smax(t) is the maximum surface settlement just above the tunnel’s center line at time t, and a and b are parameters determined by ground and tunnel conditions. According to the study of
Y. Cao et al. / Computers and Geotechnics 61 (2014) 144–152
151
Fig. 7. Comparison of settlement curves at different times.
Qu [25], in this case, a = 0.4891 and b = 0.0119, the modified Peck formula is obtained as follows:
x2 Sx ¼ Smax ðtÞexp 02 2i
ð52Þ
where i0 is the modified width parameter of the settlement curve with time, and Sx is surface settlement with a distance of x from the tunnel center line. Fig. 7 shows the settlement curves predicted by nonlinear solutions, Terzaghi solutions, the modified Peck formula (denoted as PF) and field data (t = 180, 360 d) at different times. As illustrated in Fig. 7(a and b), at the early stages of consolidation, the slow consolidation of soil around the tunnel with an impermeable lining leads to much smoother settlement curves by both consolidation theories than that for permeable lining conditions. The field measured curves and predictions by the modified Peck formula are located in the range between the settlement curves representing permeable and impermeable lining conditions. According to Fig. 7, the field measured curves and predictions by the modified Peck formula are much closer to the permeable lining conditions, which is more obvious at early stages of consolidation. The possible cause could be that the hardening of grouting concrete has not been completed, or the train loading induces a certain amount of cumulative deformation in the soft soil. Then, as shown in Fig. 7(c and d), the deviation between field data and the predictions for impermeable lining conditions decreases with time. This is attributed to the sharp decrease in the permeability
and compressibility of the soil, and the arrival of a steady flow state for the impermeable lining condition. For the impermeable lining condition, the decrease of settlement rates is very slow, and the settlements predicted by both consolidation solutions increase with a steady and small amount.
6. Conclusions By introducing the nonlinear consolidation governing equation and conformal mapping, the analytical solutions for the nonlinear consolidation of soft clay around a shield tunnel, of which sealing linings are idealized as completely permeable or completely impermeable, are derived. The comparisons between the predicted excess pore pressure, degree of consolidation, ground settlement, settlement rate by nonlinear solutions and by Terzaghi consolidation solutions are carried out. The predictions of ground settlement and the settlement curve by both consolidation theories are also compared with field data and a modified Peck formula. It can be concluded that a tunnel that acts as a drain accelerates the consolidation of soil, by introducing a new drainage boundary. At the early phases of soil consolidation, the field data are closer to completely permeable conditions. During the later stages of consolidation, with a small and slightly decreased settlement rate, the nonlinearity of soil consolidation becomes more notable, and the tendency of the field settlement behaves more similar to the impermeable case.
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However, it should be noted that the assumptions of this study, that soil and sealing lining are assumed to be homogeneous and the initial excess pore pressure is distributed from the interface of the tunnel to the ground surface in a half-space, are different from actuality. The segmental linings of shield tunnel with different types of joint leaking offer different drainage boundaries for the consolidation of surrounding soil. Naturally, it leads to different surface settlement distributions from the assumed isotropic sealing linings. The loading condition during long-term consolidation is simplified by neglecting the train loading. Further studies are needed for more realistic assumptions regarding the drainage boundaries of shield tunnel, the distribution of initial excess pore pressures, and their effects on the nonlinear consolidation of soil around a tunnel with sealing linings. Additionally, the influence of train loading should be studied in further work.
References [1] Mair RJ. Tunnelling and geotechnics: new horizons. Geotechnique 2008;58(9):695–736. [2] Peck RB. Deep excavations and tunnelling in soft ground. In: Proc. 7th int. conf. on SMFE; 1969. p. 225–90. [3] Fang YS, Lin SJ, Lin JS. Time and settlement in EPB shield tunnelling. Tunnels Tunn 1993;25(11):27–8. [4] Wongsaroj J, Soga K, Mair RJ. Modelling of long-term ground response to tunnelling under St James’s Park London. Geotechnique 2007;57(1):75–90. [5] Mair RJ, Taylor RN. Bored tunnelling in the urban environment: state-of-theart report and theme lecture. In: Proc. 14th int. on. soil mech. found. eng, vol. 4, Hamburg; 1997. p. 2353–85. [6] Ward WH, Thomas HSH. The development of earth loading and deformation in tunnel linings in London Clay. In: Proc. 6th int. conf. soil mech. found. eng, vol. 2, Toronto; 1965. p. 432–6. [7] Palmer JHL, Belshaw DJ. Deformations and pore pressure in the vicinity of a precast, segmented, concrete-lined tunnel in clay. Can Geotech J 1980;17(2): 174–84.
[8] Ward WH, Pender MJ. Tunnelling in soft ground: general report. In: Proc. 10th int. conf. soil mech. found. eng, vol. 4, Stockholm; 1981. p. 261–75. [9] Zhang DM, Huang HW, Wang JM. Numerical study on the effect of grouting on long-term settlement of tunnels in clay. In: Comput Sci-ICCS. Berlin Heidelberg: Springer; 2007. p. 1114–21. [10] Shin JH. Numerical study of the effect of groundwater movement on long-term tunnel behaviour. Geotechnique 2002;52(6):391–403. [11] Shin JH, Potts DM, Zdravkovic CL. The effect of pore-water pressure on NATM tunnel linings in decomposed granite soil. Can Geotech J 2005;42(6):1585–99. [12] Bobet A, Nam SW. Stresses around pressure tunnels with semi-permeable liners. Rock Mech Rock Eng 2007;40(3):287–315. [13] Wang MB, Wang G. A modified stress-displacement solution for a pressure tunnel with a permeable liner in an elastic porous medium based on a new model. Rock Mech Rock Eng 2013;46(2):259–68. [14] Arjnoi P. Effect of drainage conditions on porewater pore water distributions and lining stresses in drained tunnels. Tunn Undergr Space Technol 2009; 24(6):376–89. [15] Li X. Stress and displacement fields around a deep circular tunnel with partial sealing. Comput Geotech 1999;24(2):125–40. [16] Zhang DM, Ma LX, Huang HW, et al. Predicting leakage-induced settlement of shield tunnels in saturated clay. Comput Model Eng Sci (CMES) 2012;89(3): 163–88. [17] Shirlaw JN. Observed and calculated pore pressure and deformations induced by an earth balance shield: discussion. Can Geotech J 1995;32(1):181–9. [18] Duncan JM. Limitation of conventional analysis of consolidation settlement (27th Terzaghi Lecture). J Geotech Eng Div, ASCE 1993;119(9):1333–59. [19] Davis EH, Raymond GP. A non-linear theory of consolidation. Geotechnique 1965;15(2):161–73. [20] Bardon L, Berry PL. Consolidation of normally consolidated clay. J Soil Mech Found Div, ASCE 1965;91(SM5):15–35. [21] Mesri G, Rokhsar A. Theory of consolidation of clays. J Geotech Eng Div, ASCE 1974;100(GT8):889–903. [22] Xie KH, Qi T, Dong Y. Nonlinear analytical solution for one-dimensional consolidation of soft soil under cyclic loading. J Zhejiang Univ Sci A 2006;7(8): 1358–64. [23] Verruijt A. A complex variable solution for a deforming circular tunnel in an elastic half-plane. Int J Numer Anal Meth Geomech 1997;21(2):77–89. [24] Laver RG. Long-term behaviour of twin tunnels in London clay. Ph.D. thesis. Cambridge University; 2011. [25] Qu Jili. Study of ground long-term settlement induced by shield construction. TongJi University; 2002 [in Chinese].
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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Analytical solutions for nonlinear consolidation of soft soil around a shield tunnel with idealized sealing linings Yi Cao, Jun Jiang ⇑, Kang-He Xie, Wei-Ming Huang College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China
a r t i c l e
i n f o
Article history: Received 23 February 2014 Received in revised form 21 May 2014 Accepted 22 May 2014
Keywords: Shield tunnel Nonlinear consolidation Analytical solution Sealing linings
a b s t r a c t This paper presents the analytical solutions for nonlinear consolidation of soft soil around a shield tunnel with idealized sealing linings. By introducing the empirical relation between permeability and compressibility, along with the conformal transformation, the governing equations of nonlinear consolidation are established, and the corresponding analytical solutions are derived. Then, the Terzaghi consolidation solutions are derived from the degenerate governing equation of nonlinear consolidation. Through the predictions of different consolidation theories in both completely permeable and impermeable lining conditions, the influences of a tunnel acting as a drain and impacting the dissipation of pore pressure, degree of consolidation, long-term ground settlements and ground settlement rates are investigated. During the early stages of consolidation, the case studies reveal that the predictions made by this study strongly agree with the field data when a completely permeable lining is applied. This study confirms that a tunnel acting as a drain can accelerate the consolidation of soil and enlarge soil deformation due to consolidation. During long term consolidation, a notable nonlinearity of the soil consolidation is exhibited by a small and gradually decreasing settlement rate, showing agreement with the tendency of field data from the impermeable conditions. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Many theoretical and field studies have been performed on tunneling induced ground settlement, yet most of these studies are concerned with short-term ground movement induced by tunnel construction [1]. However, the ground settlement around tunnels in soft clays continuously increases for many years after construction, especially around those built in coastal areas. Peck proposed an empirical formula for tunneling settlement [2], and then, Fang et al. proposed a time dependent equation for ground settlement above a tunnel’s center line [3]. Because of the large difference between the hydraulic head inside and outside tunnel linings after the construction of a tunnel, as well as joint leaking caused by segmented construction and differential settlement, Wongsaroj et al. concluded that tunnels in soft clays actually act as drains [4], which introduces a new drainage boundary condition that leads to long-term reductions in pore pressure and associated consolidation settlements. Mair and Taylor argued that the resulting settlement profile at the ground surface will tend to be considerably wider than the profile associated with construction [5]. ⇑ Corresponding author. Tel.: +8613957115706. E-mail addresses: [email protected] (Y. Cao), [email protected] (J. Jiang), [email protected] (K.-H. Xie), [email protected] (W.-M. Huang). http://dx.doi.org/10.1016/j.compgeo.2014.05.014 0266-352X/Ó 2014 Elsevier Ltd. All rights reserved.
Further research has confirmed, through measurements of pore pressures around tunnels, that tunnels constructed in soils with low-permeability act as drainage boundaries [6,7]. Moreover, various case studies and numerical analyses reveal that segmentally lined tunnels still act as drains when grouted [8,9]. Since then, many studies on this problem have been performed by using analytical and numerical tools and by considering the lining as a homogeneous permeable body [10–16]. However, the nonlinear behavior of soil in long-term consolidation has not been taken into consideration by the aforementioned studies. Based on an enormous amount of field measurements and theoretical research work, Shirlaw concluded that the proportion of long-term consolidation induced by construction disturbance to total settlement is between 30% and 90% [17]. Considering the immediately occurring construction settlement, the proportion of consolidation to total settlement after tunnel operation is even higher. Duncan found that the permeability and compressibility of soft clays change very much during long-term consolidation, causing a serious deviation from using the conventional consolidation theory in predicting the long-term settlement of soft clays [18]. As an effective tool to overcome the limitations of the conventional consolidation theory, the nonlinear consolidation theory can well predict the nonlinear behavior of soft clays [19–22].
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This paper idealizes the linings of tunnels as being completely permeable or completely impermeable, which represents the upper and lower limits of the induced soil consolidation behavior, and investigates how a tunnel acting as a drain affects the long-term behavior of the surrounding soil. In this paper the tunnel support is assumed to be rigid, so that the ground deformation around the tunnel section is short-term behavior compared with the effects of nonlinear consolidation. It is assumed that most of the ground deformation around the tunnel section is already completed during tunneling. Thus the tunnel support is not taken into account in the analysis presented. By introducing the classic e-lg k relation of nonlinear consolidation, and the conformal transformation, the corresponding consolidation equation is established. Then, the analytical solutions are developed for a long-term nonlinear consolidation with idealized drainage boundaries. After that, the obtained nonlinear solutions for excess pore pressure, degree of consolidation, and ground settlements are discussed by comparing them with the conventional Terzaghi consolidation solution corresponding to these special cases. Through comparison with settlement curves of field measured data and a modified Peck formula, the rationality of the upper and lower limits by the presented analytical solutions of both idealized lining conditions is discussed. 2. Statement of the problem The geometry of a circular tunnel constructed in soft clay with an idealized lining is shown in Fig. 1. The radius of the tunnel buried at a depth ‘h’ in soft clay is r, and the origin of the coordinate is at the ground surface above the tunnel. Considering Davis’s hypothesis for nonlinear consolidation of soil [19], the basic assumptions made in developing the solutions presented in this study are stated below: 1. The longitudinal length of the tunnel is large enough to meet plane strain conditions. 2. The soil around the tunnel is regarded as a half space with a cavity. 3. The soil around the tunnel is an isotropic saturated porous medium, and the permeability and compressibility coefficients are subjected to the e-lg k and e-lg r relation, respectively, during the consolidation. 4. Soil particles and pore water are incompressible. Darcy’s law is valid. The excess pore pressure at the ground surface and at an infinite depth is zero. 5. Soil deformation during consolidation is small, neglecting the effect on the coordinate system.
6. Soil deforms freely, neglecting the arch effect of soil itself and the tunnel. 7. The construction of a shield tunnel only causes excess pore pressure, and does not change the distribution of total vertical stress. 3. Solutions for nonlinear consolidation of soft clay around tunnel 3.1. Governing equation Based on basic assumption (3), the permeability and compressibility coefficients of the soil are respectively subjected to an empirical nonlinear relation by Davis, Bardon, Mesri, and Xie [19–22], as below:
0 r e ¼ e0 C c lg 0 r0 ks e ¼ e0 þ C k lg ks0
ð1Þ ð2Þ
where e is the void ratio, e0 is the initial void ratio, r00 is the initial effective stress, r0 is the effective stress, ks0 is the initial permeability coefficient of the soil, ks is the permeability coefficient of the soil, Cc is the initial compression index, and Ck is the initial permeability index. Then, from Eqs. (1) and (2), it can be obtained that
ks ¼ ks0
r00 =r0
CCc
k
1 @e r0 ¼ mm0 00 mm ¼ 1 þ e0 @ r0 r
ð3Þ ð4Þ
where mv is the coefficient of compressibility, and mm0 ¼ C c =ð1 þ e0 Þr00 ln 10 is the initial coefficient of compressibility. Then, substituting mv0 into Eq. (4), we can obtain
mm ¼
Cc 1 ð1 þ e0 Þ ln 10 r0
ð5Þ
From the free strain assumption, the continuity equation of saturated soil can be derived
1 @ @ Du 1 @ @ Du @ em þ ¼ ks ks @x @y cw @x cw @y @t @ em 1 @e ¼ 1 þ e0 @t @t
ð6Þ ð7Þ
where cw is the bulk density of water, Du is the excess pore pressure, and ev is the body strain. From the principle of effective stress, we can obtain
@ r0 @ Du ¼ @t @t
ð8Þ
Then, substituting Eqs. (5) and (8) into Eq. (7), we can obtain
@ em Cc 1 @ Du ¼ @t ð1 þ e0 Þ ln 10 r0 @t
ð9Þ
Assuming that the compressibility and permeability are decreasing all along with increasing pressure, (as done by Davis) [19], i.e., Cc/Ck = 1, by substituting Eqs. (3) and (9) into Eq. (6), the governing equation is then derived as
0 0 1 @ r @ Du 1 @ r @ Du þ ks0 00 ks0 00 cw @x r @x cw @y r @y ¼ Fig. 1. Geometry of a tunnel in soft clay.
Cc 1 @ Du ð1 þ e0 Þ ln 10 r0 @t
Using the substitution w:
ð10Þ
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w ¼ ln
r0
Du0 þ r00
ð11Þ
ffi Duc jpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þðHyÞ2 PH
where u0 is the initial excess pore pressure. Then, the governing equation can be simplified as:
! @ w @ w @w þ 2 ¼ @x2 @y @t 2
Cm
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h rs 2ðq0 cos hÞ 1þq2 2q cos h ; i:e:; Duc ¼ q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hr 2ðq0 cos hÞ 1þR2 2R cos h ¼0 Duc jqPq0 ¼ 0
ffi ¼q Duc jr6pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þðHyÞ2 6H
ð16Þ
2
ð12Þ
where Cm = ks0/(cwmm0) is the coefficient of consolidation.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where rs is x2 þ ðh yÞ , and q0 is the boundary of the distribution of excess pore pressure in the complex plane. The drainage boundary condition of the interface between the tunnel and the surrounding soil can be idealized in two forms due to different lining conditions:
3.2. Complex conformal transformation Because of the difficulty in analyzing the nonlinear consolidation of soft clay surrounding a tunnel in rectangular coordinate system, the considered region in the physical plane Z = x + yi is mapped to a circular ring region in the image plane f by using the complex conformal transformation, where the outer boundary with radius 1 of the ring region is the ground surface in the Z plane and the inner boundary with radius R of the ring region is the interface between the tunnel and soil, as shown in Fig. 2. The conformal mapping function by Verruijt [23] is given as below:
z ¼ xðfÞ ¼ ih
1 R2 1 þ f 1 þ R2 1 f
ð13Þ
where h is the tunnel depth, and R is the radius of the tunnel in image plane f, in which
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 r R ¼ h h r2
ð14Þ
Then, the governing equation after conformal mapping can be obtained:
Cm
! @2w 4a2 @w þ 2 ¼ 2 2 @ @t g 2 @n 2 ð1 nÞ þ g
@2w
where a ¼
ð15Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h r2 .
3.3. Boundary and initial conditions According to basic assumption (4), the excess pore pressure at the ground surface and at an infinite depth is zero. The magnitude of the initial excess pore pressure at the interface between the tunnel and soil is known as q, and the distribution of excess pore pressure by the tunnel construction is assumed to be approximately uniform around the tunnel hoop and linear along the radial direction. Then, the initial distribution of the excess pore pressure from tunnel construction Duc, can be described as follows:
(1) Completely permeable, i.e., pore pressure is zero at the interface between the lining and the soil. (2) Completely impermeable. When completely permeable lining is applied, the shield tunnel introduces a new drainage boundary for soil consolidation, which strongly affects the dissipation of pore pressure. When the final steady flow state is reached, the decrease of hydrostatic pore pressure in the surrounding soil, along with the dissipation of excess pore pressure by the tunnel construction, defines the incremental effective stress, which is stated as follows:
Du0 ¼ Duc þ u0 us ¼ q
h rs þ cw y us hr
ð17Þ
where u0 is the hydrostatic pore pressure before the tunnel construction, us is the pore pressure when a steady flowing state is reached, and rs is the distance of calculation point from tunnel center. When tunnel lining is assumed to be completely permeable, the pore pressure at the soil-tunnel lining interface is zero, and the total water head is just the position hydraulic water head. According to the analytical solution of lining flow and soil flow by Laver [24]:
Qt ¼
2pr s ks us ujq¼R
ð18Þ
cw ðrs r2 Þ
where Qt is the lining flow per unit tunnel length per unit time, and r2 is the outer radius of the tunnel. Under the condition of completely permeable lining, u|q=R, the pore pressure at the soil-tunnel interface is zero. Soil flow per unit tunnel length per unit time along the flowing interface of which the distance from the tunnel lining is rs r2, can be given as
Qs ¼
2pks ½hcw us s cw ln 1 þ hr rs
ð19Þ
Given that the water flow through different seepage interfaces in the soil going into the tunnel are equal, the distribution of pore pressures of a steady flow state can be derived from Eqs. (18) and (19) as follows:
ðr s r 2 Þ us ¼ hcw sÞ r s ln 1 þ ðhr þ ðr s r 2 Þ rs
ð20Þ
Then, the 1st idealized boundary (completely permeable) with the initial conditions can be respectively described in the Z plane and f plane as
Dujt¼0 ¼ Du0 Dujq¼1 ¼ 0 ; i:e:; Dujq¼R ¼ 0 Dujt¼1 ¼ 0 Fig. 2. The region after conformal mapping.
wðq; h; 0Þ ¼ ln
r00
Du0 þr00
wð1; h; tÞ ¼ 0
0
wjq¼R ¼ ln Du0rþr0
0
wjt¼1 ¼ 0
q¼R
¼0
ð21Þ
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Y. Cao et al. / Computers and Geotechnics 61 (2014) 144–152
Then, the 2nd idealized boundary (completely impermeable) with the initial conditions can be respectively described in the Z plane and f plane as
Dujt¼0 ¼ Du0 Dujq¼1 ¼ 0
; i:e: @ Du
¼0 @q
wðq; h; 0Þ ¼ ln
Dujt¼1 ¼ 0
wjt¼1 ¼ 0
q¼R
Du0 þr00
wð1; h; tÞ ¼ 0
@w
@ Du
¼ @w @q
@u @ q
q¼R
r00
Then, substituting the characteristic value kn into Eq. (22), the following can be derived:
@T n C m k2n þ Tn ¼ 0 @t 4a2
ð30Þ
By using the Laplace transform, we can get
q¼R
¼0
ð22Þ Tn ¼ e
C m k2 nt 4a2
ð31Þ
Then, we can obtain
wðq; h; tÞ ¼ Wðq; hÞTðtÞ ¼
3.4. The derivation of the general solution
1 X W nT n
ð32Þ
0
From the governing equation and boundary conditions stated above, using the method of separation of variables, i.e., w(q, h, t) = W(q, h)T(t), the governing equation can be transformed to:
( @ 2 W 1 @W 1 @ 2 W k2 @T C m k2 þ þ 2 þ W ¼0 þ 2 T ¼0 2 2 2 @q @t 4a q @ q q @h ð1 þ q2 2q cos hÞ ð23Þ where k is the characteristic value, and using the substitution pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ¼ 1= 1 þ q2 2q cos h, we can obtain:
From Eq. (11), the excess pore pressure can be obtained:
Duðq; h; tÞ ¼ ðDu0 þ r00 Þð1 ew Þ
ð33Þ
3.6. The analytical solution for the 2nd idealized boundary condition After separation of variables, the 2nd idealized boundary condition is
Wjq¼1 ¼ 0
¼0
ð34Þ
@W
@q
q¼R
@ 2 W 1 @W þ þ k2 W ¼ 0 @ v2 v @ v
ð24Þ
The general solution of Eq. (24) is
WðvÞ ¼ AJ 0 ðkvÞ þ BN0 ðkvÞ
ð25Þ
where J0(kx) is a zero-order Bessel function, N0(kx) is a zero-order Neumann function, and A and B are undetermined coefficients of the boundary conditions.
Substituting Eq. (34) into general solution (25), the characteristic equation for the 2nd idealized boundary condition can be obtained:
k k
J pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffi N0 pffiffiffiffiffiffiffiffiffiffiffiffiffi
0 22 cos h 22 cos h
¼0
J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
N
1 1 2 2 1þR 2R cos h
W n ðq
After the separation of variables, the 1st idealized boundary condition is
ð26Þ
Substituting Eq. (26) into general solution (25), the following equations can be obtained
8 k k ffi þ BN 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼0 > < AJ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi 22 cos h 22 cos h > k k : AJ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi þ BN 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ¼0 2 2 1þR 2R cos h
3 kn ! J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1 6 7 k k 1þR2 2Rcos h n n N 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ¼ Gn 6 4J0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ q2 2q cos h 1 þ q2 2q cos h 5 kn N pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
; hÞ ¼ Gn W 0n
Wjq¼R ¼ 0
1þR 2R cos h
Then, we can obtain the general solution as:
3.5. The analytical solution for the 1st idealized boundary condition
Wjq¼1 ¼ 0
ð35Þ
1
1þR2 2Rcos h
ð36Þ
where Gn is an undetermined coefficient defined by the boundary conditions. Similarly, we can obtain
Tn ¼ e
C m k2 nt 4a2
ð37Þ
Then, the excess pore pressure can be obtained
ð27Þ
Duðq; h; tÞ ¼ ðDu0 þ r00 Þð1 ew Þ
ð38Þ
1þR 2R cos h
Considering Eq. (26), k needs to satisfy the following characteristic equation to obtain nonzero solutions of A and B:
3.7. The solution for the undetermined coefficient Gn
k k
J pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffi N0 pffiffiffiffiffiffiffiffiffiffiffiffiffi
0 22 cos h 22 cos h
¼ 0
J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k ffi ffi
N0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0 2 2
According to the nature of the Bessel function, the characteristic equations with different characteristic values are weighted orthogonally in the interval of [R, 1]:
1þR 2R cos h
ð28Þ
Z
1þR 2R cos h
The characteristic equation is denoted as f(k) = 0, which is an even function of k, ignoring these negative roots. After gradually searching its arranged roots sequence fk1 ; k2 ; k3 kn g from zero, the general solution can be obtained as 3 kn ! J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 0 6 7 k k 1þR2 2R cos h n n N0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 ¼ Gn 6 4J0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ q2 2q cos h 1 þ q2 2q cosh 5 kn N pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
; hÞ ¼ Gn W 0n
W n ðq
0
1þR2 2R cosh
ð29Þ
where Gn is an undetermined coefficient determined by boundary conditions.
1
W n ðvÞW m ðvÞvdv ¼ 0 n – m
ð39Þ
R
By the initial condition
0 Du0 ¼ Duðq; h; 0Þ ¼ ðDu0 þ r00 Þ 1 eRGn W n T n ð0Þ 0 ¼ ðDu0 þ r00 Þ 1 eRGn W n ðvÞ
ð40Þ
We can obtain:
1 X r00 Gn W 0n ðvÞTð0Þ ¼ ln Du0 þ r00 1
ð41Þ
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Y. Cao et al. / Computers and Geotechnics 61 (2014) 144–152
where Gn is as stated below:
From the nature of the Bessel function, we can obtain
ln Du0 þr0 W 0n ðvÞvdv R 0 Gn ¼ R 1 02 W n ðvÞvdv R R 1 r00 0 qcos h ln Du0 þr0 W n ðvÞ 2 dq R 0 ð1þq2 2q cos hÞ ¼ R 1 02 qcos h W n ðvÞ 2 dq R ð1þq2 2q cos hÞ R1
r00
R1 Gn ¼
ð42Þ
3.8. The average degree of consolidation according to the excess pore pressure The average degree of consolidation according to the excess pore pressure, Up, is given by
R 2p R 1
rðDu0 Dun Þdrdh R R 2p R 1 r Du0 drdh R 0 R 2p R 1 rðDu0 þ r0 Þð1 ew Þdrdh ¼ 1 0 R R 2p R 1 0 r Du0 drdh R 0 0
Up ¼
ð43Þ
3.9. The solution for consolidation ground settlement The volumetric strain of soil is given by
em ¼ ðe0 eÞ=ð1 þ e0 Þ
ð44Þ
Substituting Eq. (1) into Eq. (44), we obtain
em ¼
0 0 Cc r Cc r þ Du0 Du lg 0 ¼ lg 0 1 þ e0 r0 1 þ e0 r00
ð45Þ
From the generalized Hooke’s Law and plane strain assumption, the vertical strain of soil at any particular time and point can be obtained as
K 0 þ ð1 mÞ ey ðx; y; tÞ ¼ em ð1 2mÞðK 0 þ 1Þ
ð46Þ
where K0 is the coefficient of earth pressure at rest, and t is Poisson’s ratio of the soil. Then, the surface settlement S(t) above the tunnel at any time t and distance from the tunnel center line x can be obtained:
SðtÞ ¼
Z
0
ey ðx; y; tÞdy
ð47Þ
ðhþrÞ
4. Degeneration of the solution If basic assumption (3) is changed following the Terzaghi consolidation theory, it can be stated as ‘‘the soil around the tunnel is an isotropic saturated porous medium, and the permeability and compressibility coefficients of the soil surrounding the tunnel are constant during consolidation.’’ Then, the nonlinear consolidation governing Eq. (10) is degenerate to Terzaghi’s consolidation governing equation as shown below:
cm
@ 2 Du @ 2 Du þ @x2 @y2
! ¼
@ Du @t
ð48Þ
where cm = k/(cwmm0). When the same boundary and initial conditions, as well as conformal mapping, are applied, the corresponding analytic solutions for two idealized sealing linings can be derived as:
Duðq; h; tÞ ¼
1 X Gn W n T n n¼1
ð49Þ
Du0 W n ðvÞvdv ¼ R1 2 W n ðvÞvdv R
R
R1
qcos h Du0 W n ðvÞ 2 dq ð1þq2 2q cos hÞ R1 2 qcos h W n ðvÞ 2 dq R 2
R
ð50Þ
ð1þq 2q cos hÞ
The corresponding analytic solutions for degree of consolidation and ground settlement can be derived from the former statement. 5. Prediction of long-term behavior of soil around a shield tunnel To investigate the rationality of the presented solutions, the predictions of the presented analytic solutions are compared with the corresponding solutions of Terzaghi’s consolidation theory and field data from Shanghai Metro Line No. 1 by Qu [25]. The length of Shanghai Metro Line No. 1 is 14.6 km, the diameter of the tunnel is 6.2 m, and the thickness of overlay soil ranges from 6 m to 8 m. The tunnel is located in saturated mucky clay, which is a typical soft clay. At the calculated cross section, the tunnel center is located at the depth of 11 m, around which the soft clay layers are equivalent to a homogeneous soil with an initial permeability coefficient k0 = 2.1 109 ms1, an initial compression index Cc = 0.25, an initial void ratio e0 = 1, a Poisson’s ratio of soil t = 0.36, and a coefficient of earth pressure at rest K0 = 0.43. The solutions of nonlinear consolidation of soft clay around the tunnel with idealized completely permeable and completely impermeable lining for excess pore pressure, degree of consolidation, settlement rates, etc. are compared with the corresponding solutions of Terzaghi’s consolidation theory. The predictions of ground settlement by nonlinear and Terzaghi consolidation theories are compared with the field data. The predictions are denoted as below: (A) Solutions of Terzaghi consolidation when the lining is idealized as completely permeable, PT. (B) Solutions of Terzaghi consolidation when the lining is idealized as completely impermeable, IPT. (C) Solutions of presented nonlinear consolidation when the lining is idealized as completely permeable, PN. (D) Solutions of presented nonlinear consolidation when the lining is idealized as completely impermeable, IPN. (E) Pore pressure when steady flow state is reached, when the tunnel lining is completely permeable, SFPP. (F) Hydrostatic Pore Pressure, HPP. (G) Field data of ground settlement from Shanghai Metro Line No. 1, FS. 5.1. Dissipation of excess pore pressure The predictions of excess pore pressure varying with depth above the center line of the tunnel by the presented nonlinear consolidation solutions and Terzaghi consolidation solutions with idealized linings are shown in Fig. 3(a–d). Fig. 3 illustrates that whether permeable or impermeable lining conditions are applied, the predicted dissipation of the excess pore pressure by nonlinear consolidation is slower than that of Terzaghi consolidation. Along with the dissipation of excess pore pressure and increase of effective stress, the changes in the soil permeability and compressibility in these two consolidation theories are very different. The sealing condition determines the reduction of pore pressure during consolidation. The reduction of pore pressure when lining is completely impermeable is just the excess pore pressure due to tunnel construction, i.e., Duc, while the reduction of pore pressure when lining is completely permeable is Duc + HPPSFPP.
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At the early stages of consolidation, if the completely impermeable lining condition is applied, the dissipations of pore pressure by both consolidation theories are relatively closer than those of the completely permeable lining condition. Then, the dissipation of pore pressure by Terzaghi consolidation is much quicker than that by nonlinear consolidation, which means that the deviation between Terzaghi and the presented solutions increases with time. This deviation can be explained by a decrease in the soil permeability and compressibility, which slow down the dissipation of pore pressure. However, in the Terzaghi consolidation, the soil permeability and compressibility is constant. On the contrary, at the early stage of consolidation (t = 0– 100 d), when the completely permeable lining condition is applied, the dissipations of pore pressure by Terzaghi consolidation is also quicker than nonlinear consolidation, but the deviation of pore pressure by these two consolidation theories decrease with time. At t = 400 d, the consolidation of soil around the tunnel is almost complete, the excess pore pressure from both consolidation theories has dissipated by approximately 90%. It can be concluded that
the tunnel acting as a drain accelerates the dissipation of pore pressure. The closer to the completion of consolidation, the smaller the deviation between the pore pressures predicted by both consolidation theories is. 5.2. Growth of the degree of consolidation The predictions of the average degree of consolidation with time by the presented nonlinear consolidation solutions and Terzaghi consolidation solutions with idealized linings are shown in Fig. 4. Then, it can be similarly concluded from Fig. 4, as from Fig. 3 (a–d), that at the early phases of consolidation of soil around the tunnel with an impermeable lining, the growth of the two consolidation theories are very close. In the analysis, with decreasing compressibility and permeability of soil, the difference between the corresponding average degree of consolidation by nonlinear consolidation and Terzaghi consolidation reaches its maximum at t = 400–600 d. When tunnel lining is completely permeable, and
Pore pressure, u (kPa) 50
100
200
PT--Pore pressure in Permeable lining by Terzaghi theory IPT--Pore pressure in imPermeable lining by Terzaghi theory PN--Pore pressure in Permeable lining by nonlinear theory IPN-- Pore pressure in imPermeable lining by nonlinear theory SFPP-- Pore pressure when steady flowing state HPP-- Hydrostatic Pore Pressure
5
depth, z (m)
Pore pressure, u (kPa)
150
10
tunnel
0 0
15
20
50
200
PT--Pore pressure in Permeable lining by Terzaghi theory IPT--Pore pressure in imPermeable lining by Terzaghi theory PN--Pore pressure in Permeable lining by nonlinear theory IPN-- Pore pressure in imPermeable lining by nonlinear theory SFPP-- Pore pressure when steady flowing state HPP-Hydrostatic Pore Pressure
10
tunnel
50d
(a)
20
(c)
300d
Pore pressure, u (kPa) 100
Pore pressure, u (kPa)
150
PT--Pore pressure in Permeable lining by Terzaghi theory IPT--Pore pressure in imPermeable lining by Terzaghi theory PN--Pore pressure in Permeable lining by nonlinear theory IPN-- Pore pressure in imPermeable lining by nonlinear theory SFPP-- Pore pressure when steady flowing state HPP-- Hydrostatic Pore Pressure
10
tunnel
0 0
200
50
100
10
tunnel
15
15
100d
(b)
20
150
200
PT--Pore pressure in Permeable lining by Terzaghi theory IPT--Pore pressure in imPermeable lining by Terzaghi theory PN--Pore pressure in Permeable lining by nonlinear theory IPN-- Pore pressure in imPermeable lining by nonlinear theory SFPP-- Pore pressure when steady flowing state HPP-- Hydrostatic Pore Pressure
5
depth, z (m)
50
5
depth, z (m)
150
15
0 0
20
100
5
depth, z (m)
0 0
(d)
Fig. 3. Distribution of pore pressure along depth.
600d
150
Y. Cao et al. / Computers and Geotechnics 61 (2014) 144–152
considering the whole time period of consolidation, the growth of the predicted average degree of consolidation by Terzaghi’s solution is faster than that by the presented nonlinear solution. The curves are very close when t = 400 d. This result illustrates that when a tunnel acts as a drain, it accelerates the consolidation of soil significantly. 5.3. Ground settlement and settlement rate at the tunnel center line due to soil consolidation with time The predicted ground settlement and settlement rate by presented nonlinear consolidation solutions and Terzaghi consolidation solutions with idealized linings, and field data of ground settlement from Shanghai Metro Line No. 1 are shown in Figs. 5 and 6, respectively. It is shown in Fig. 5 that the predicted ground settlement of the tunnel with a completely permeable lining is much larger than that of the tunnel with a completely impermeable lining, using either the presented nonlinear consolidation solution or the Terzaghi consolidation solution. According to Eqs. (45) and (47), the dissipation of pore pressure determines the soil deformation due to consolidation. Together with Eq. (17), it is concluded that tunnel acting as a drain enlarged the soil deformation due to consolidation. Moreover, this result illustrates that at the early stages of consolidation, the predicted ground settlements by both consolidation solutions in the permeable lining condition are very close to the field data from Shanghai Metro Line No. 1, revealing that at the early stages of soil consolidation, the completely permeable lining condition is closer to the real drainage condition. After approximately 100 d of consolidation, the settlement rate of field data decreased, with a tendency of ground settlement being closer to the predictions by the presented nonlinear solution and Terzaghi consolidation solution in the impermeable lining condition. At this phase, the field measured ground settlements were between the predictions of the two idealized sealing linings, with a curve closer to the permeable lining condition. Fig. 6 reveals that at the early stages of consolidation (approximately 0–150 d), the predicted settlements rates in completely permeable lining conditions by either of these consolidation theories are much larger than in impermeable lining conditions. Similar to the conclusion from Figs. 3 and 4, the tunnel acts as a drain, accelerating the consolidation of soil around the tunnel. During the soil consolidation in permeable lining conditions, there is a rapid decrease in the predicted settlement rate by both of these consolidation theories. In impermeable lining conditions, the curves of the settlement rates are smoother, indicating a much slower decrease, while the prediction by nonlinear consolidation
Fig. 6. Settlement rates due to soil consolidation around the tunnel with idealized linings.
0.8
has a smaller decrease than that given by Terzaghi consolidation. After consolidation of approximately 200 d, the predicted settlement rates by the nonlinear consolidation solution are larger than those of Terzaghi consolidation solution. According to the discussion above, when the same drainage boundary condition is applied during the first stage of consolidation (t < 100 d), the deviation between the predictions of ground settlement by nonlinear solution and the Terzaghi solution is not that significant. Then, the deviation increases after the first stage of consolidation. When consolidation is nearly finished, the deviation decreases. It is noted that the deviation for the permeable lining condition is much smaller than that in impermeable lining conditions.
0.6
5.4. Settlement curve with time
1.0
Average degree of consolidation, U
Fig. 5. Ground settlements due to soil consolidation around the tunnel with idealized linings.
The hyperbolic model for time dependent ground settlement above the tunnel center line proposed by Fang et al. [3] is shown as follows:
0.4 PT IPT PN IPN
0.2 0.0
0
200
400
600
800
time, t (day) Fig. 4. Growth of the degree of consolidation with time.
Smax ðtÞ ¼ 1000
t a þ bt
ð51Þ
where Smax(t) is the maximum surface settlement just above the tunnel’s center line at time t, and a and b are parameters determined by ground and tunnel conditions. According to the study of
Y. Cao et al. / Computers and Geotechnics 61 (2014) 144–152
151
Fig. 7. Comparison of settlement curves at different times.
Qu [25], in this case, a = 0.4891 and b = 0.0119, the modified Peck formula is obtained as follows:
x2 Sx ¼ Smax ðtÞexp 02 2i
ð52Þ
where i0 is the modified width parameter of the settlement curve with time, and Sx is surface settlement with a distance of x from the tunnel center line. Fig. 7 shows the settlement curves predicted by nonlinear solutions, Terzaghi solutions, the modified Peck formula (denoted as PF) and field data (t = 180, 360 d) at different times. As illustrated in Fig. 7(a and b), at the early stages of consolidation, the slow consolidation of soil around the tunnel with an impermeable lining leads to much smoother settlement curves by both consolidation theories than that for permeable lining conditions. The field measured curves and predictions by the modified Peck formula are located in the range between the settlement curves representing permeable and impermeable lining conditions. According to Fig. 7, the field measured curves and predictions by the modified Peck formula are much closer to the permeable lining conditions, which is more obvious at early stages of consolidation. The possible cause could be that the hardening of grouting concrete has not been completed, or the train loading induces a certain amount of cumulative deformation in the soft soil. Then, as shown in Fig. 7(c and d), the deviation between field data and the predictions for impermeable lining conditions decreases with time. This is attributed to the sharp decrease in the permeability
and compressibility of the soil, and the arrival of a steady flow state for the impermeable lining condition. For the impermeable lining condition, the decrease of settlement rates is very slow, and the settlements predicted by both consolidation solutions increase with a steady and small amount.
6. Conclusions By introducing the nonlinear consolidation governing equation and conformal mapping, the analytical solutions for the nonlinear consolidation of soft clay around a shield tunnel, of which sealing linings are idealized as completely permeable or completely impermeable, are derived. The comparisons between the predicted excess pore pressure, degree of consolidation, ground settlement, settlement rate by nonlinear solutions and by Terzaghi consolidation solutions are carried out. The predictions of ground settlement and the settlement curve by both consolidation theories are also compared with field data and a modified Peck formula. It can be concluded that a tunnel that acts as a drain accelerates the consolidation of soil, by introducing a new drainage boundary. At the early phases of soil consolidation, the field data are closer to completely permeable conditions. During the later stages of consolidation, with a small and slightly decreased settlement rate, the nonlinearity of soil consolidation becomes more notable, and the tendency of the field settlement behaves more similar to the impermeable case.
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However, it should be noted that the assumptions of this study, that soil and sealing lining are assumed to be homogeneous and the initial excess pore pressure is distributed from the interface of the tunnel to the ground surface in a half-space, are different from actuality. The segmental linings of shield tunnel with different types of joint leaking offer different drainage boundaries for the consolidation of surrounding soil. Naturally, it leads to different surface settlement distributions from the assumed isotropic sealing linings. The loading condition during long-term consolidation is simplified by neglecting the train loading. Further studies are needed for more realistic assumptions regarding the drainage boundaries of shield tunnel, the distribution of initial excess pore pressures, and their effects on the nonlinear consolidation of soil around a tunnel with sealing linings. Additionally, the influence of train loading should be studied in further work.
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