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The ground reaction curve of underwater tunnels considering seepage forces
Tunnelling and Underground Space Technology 25 (2010) 315–324
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The ground reaction curve of underwater tunnels considering seepage forces Young-Jin Shin a, Byoung-Min Kim b, Jong-Ho Shin c, In-Mo Lee d,* a
Civil Works Division, Samsung C&T, Seoul, Republic of Korea Underground Space Construction Technology Centre, Seoul, Republic of Korea c Department of Civil Engineering, Konkuk University, Seoul, Republic of Korea d Department of Civil Engineering, Korea University, Seoul, Republic of Korea b
a r t i c l e
i n f o
Article history: Received 17 November 2008 Received in revised form 28 December 2009 Accepted 6 January 2010 Available online 18 February 2010 Keywords: Tunnel Seepage forces Ground reaction curve Theoretical solution
a b s t r a c t When a tunnel is excavated below groundwater table, the groundwater flows into the excavated wall of tunnel and seepage forces are acting on the tunnel wall. Such seepage forces significantly affect the ground reaction curve which is defined as the relationship between internal pressure and radial displacement of tunnel wall. In this paper, seepage forces arising from the ground water flow into a tunnel were estimated quantitatively. Magnitude of seepage forces was estimated based on hydraulic gradient distribution around tunnel. Using these results, the theoretical solutions of ground reaction curve with consideration of seepage forces under steady-state flow were derived. A no-support condition was taken into account. The theoretical solution derived in this study was validated by numerical analysis. The changes in the ground reaction curve according to various ground and groundwater conditions were investigated. Based on the results, the application limit of theoretical solution was suggested. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction When a tunnel is excavated below groundwater table, the groundwater may flow into the tunnel, and consequently seepage forces will occur through the ground. This may seriously influence on the behavior of the tunnel. The ground behavior due to tunneling can be understood theoretically by the convergence-confinement method. The convergence-confinement method is based on the principle in which a tunnel is stabilized by controlling its displacements after installation of the support near the tunnel face. The convergence-confinement method consists of three elements: the longitudinal deformation profile, the ground reaction curve, and the support characteristic curve. The longitudinal deformation profile, based on the assumption of nosupport, shows the radial displacement of the tunnel cross-section in the longitudinal direction from the tunnel face. The support characteristic curve describes the increasing pressure that acts on the supports as the radial displacement increases. Lastly, the ground reaction curve shows the increasing trends of radial displacement as the internal pressure of the tunnel decreases. Tunneling under ground water table induces additional seepage stress (Shin et al., submitted for publication) and the seepage forces are likely to have a strong influence on the ground reaction curve.
* Corresponding author. Tel.: +82 2 3290 3314; fax: +82 2 928 7656. E-mail address: [email protected] (I.-M. Lee). 0886-7798/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2010.01.005
The researches on ground reaction curve by Brown et al. (1983), Stille et al. (1989), Wang (1994), Carranza-Torres and Fairhurst (1999, 2000), Carranza-Torres (2002), Sharan (2003), and Oreste (2003) were studied without considering the seepage forces; they were only for the dry condition. The studies about the effects of seepage forces on the tunnel face or the support system by Muir Wood (1975), Curtis et al. (1976), Atkinson and Mair (1983), Schweiger et al. (1991), Fernandez and Alvarez (1994), Fernandez (1994), Lee and Nam (2001), Bobet (2003), Shin et al. (2005) were conducted. An elastic–brittle–plastic strain analysis of displacements around circular openings in an isotropic Hoek–Brown rock was performed by Sharan (2005) and the effect of water pressure on the mechanical response of cylindrical lined tunnels in elastoplastic media was investigated by Carranza-Torres and Zhao (2009). A simplified analytical solution of the ground reaction curve was suggested by Lee et al. (2007); however, mathematical solutions of ground reaction curves influenced by seepage forces have not yet been suggested. In this study, based on these previous studies, seepage forces due to tunneling under groundwater table were estimated quantitatively, and the theoretical solution of ground reaction curve considering seepage forces due to groundwater flow under steady-state flow was derived. The studies were performed for a no-support. The validity of the derived theoretical solution was examined by numerical analysis. The changes in the ground reaction curve according to various ground and ground water table conditions were investigated. Based on the results, the application limit of theoretical solution was suggested.
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2. Calculation of seepage forces 2.1. The range for calculating seepage forces
flow line
When groundwater flows into a tunnel below the groundwater table, the stress of the soil-mass would be the state of ‘‘the submerged weight plus seepage force”. In this case, seepage forces are considered one of the most influencing factors, and the calculation of seepage forces is crucial in the tunnel behavior. Seepage forces per unit volume can be formulated as shown in the following equation:
S:F=v ol: ¼
izcw A ¼ icw zA
iθ = i × sin α
ir
α
iθ
i
ð1Þ
where the hydraulic gradient, i is defined as the total head loss, Dh divided by the distance between any two points, cw is unit weight of water, A is area, and z is depth. Thus, seepage forces can be easily calculated if the total head loss, Dh is obtained at any two points. In this case, seepage forces can be calculated by obtaining the total head loss from the potential failure plane to the drainage region, and the direction of the seepage force is equal to that of flow line. In a tunnel, although seepage forces develop, the plastic region may or may not be encountered. Thus it is difficult to calculate seepage forces influencing a tunnel behavior quantitatively by using the method that was used in calculating seepage forces acting on the earth retaining structure and tunnel face. Therefore, the region for calculating seepage forces should be justified in advance to evaluate seepage forces in a tunnel. While tunneling under steady-state water flow, the groundwater flows into tunnel and the flow net as shown in Fig. 1 develops. The hydraulic gradients touching all the flow lines can be divided into a radial component of hydraulic gradient, ir and a tangential component of hydraulic gradient, ih with the angle, a as shown in Fig. 2. The radial component of hydraulic gradient, ir is used to reflect the seepage effect of tunneling in this study. This is because the radial component of the groundwater flow affects most of the displacement of tunnel wall, and the seepage forces can be estimated in both of the radial and non-radial flows by using radial component hydraulic gradient. All flows induced by tunneling under the groundwater table affect seepage forces. Therefore, the distance where the radial component of hydraulic gradient becomes zero is defined as the range for calculating seepage forces under steady-state groundwater flow, as shown in Fig. 3. Lee et al. (2007) define the range for calculating seepage forces until the critical hydraulic gradient, ir = 1 while this study extends to ir = 0.
i
ir = i × cos α
i
Fig. 2. The radial and tangential component of hydraulic gradient.
i α
i
α
r
ir = i × cos α
ir
Critical hydraulic gradient
ir = 1
Region for calculating seepage force
ir = 0
i
r
Fig. 3. The range for calculating seepage forces.
2.2. Calculation of seepage forces
Fig. 1. Flow net while tunneling under steady-state water flow.
If the lining is fully drained, ground water will seep toward the tunnel, and the seepage stress, r0i will increase due to the drag of water flowing past the soil grains. It acts in the direction of flow. Atkinson and Mair (1983) propose Eq. (2) as follows:
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dr0i ¼ cw i dS
ð2Þ
For Atkinson’s equation, r0i indicates seepage forces per volume, i is hydraulic gradient, and S is distance. If groundwater level is not varied and only flows into the radial direction, then Eq. (2) can be rewritten as following:
r0i ¼ cw
Z
ir dr
ð3Þ
where ir is the radial component of hydraulic gradient, and r is the radial distance. For the cases in which the cover depth of tunnel, C are 2.5 times, 5 times, 7.5 times and 10 times diameter of tunnel, parametric studies to investigate seepage forces are conducted with the variation of ground water height, H from 2.5 times to 10 times diameter of tunnel in a tunnel as shown like Fig. 4. The tunnel diameter remains 5 m in these parametric studies. The region for calculating seepage forces described above is shown in Table 1, the seepage forces obtained by using the above method are summarized in Table 2. The seepage pressure ratio in Table 2 is determined by generalizing seepage forces with hydrostatic pressure at the tunnel crown in a hydrostatic condition. Fig. 5 illustrates variation of seepage pressure as a function of ground water level with constant cover depth. As shown in Fig. 5, seepage pressure increases as the groundwater level increases. Fig. 6 shows the seepage pressure ratio. If the groundwater level is below the ground surface, though the seepage force acting on the tunnel is different depending on the groundwater table, the seepage pressure ratios show the range of 85–90% of the hydrostatic pressure at the tunnel crown. If the groundwater level is higher than the ground surface, the seepage pressure ratios are over 95%. Through several conducted studies, there have been different proposed ranges regarding quantification of the seepage force.
Schweiger et al. (1991) state that the magnitude of seepage force is 50–80% of the hydrostatic pressure. Lee and Nam (2001) expect that the magnitude of seepage force is 30–50% of hydrostatic pressure and Lee et al. (2007) propose it to be 50–80% of hydrostatic pressure. The differences are originated by different analytical methods. Schweiger et al. (1991) consider the deformation of ground and only first lining in calculating seepage forces. Thus, the estimated seepage force was less than the hydrostatic pressure. On the other hand, Lee and Nam (2001) consider secondary lining and a cushion effect created by drainage material in addition to the first lining. This is why the seepage force found by Lee and Nam is smaller than that found by Schweiger. Lee et al. (2007) propose a simple method for calculating seepage force by introducing the concept ‘‘critical hydraulic gradient”. The seepage forces evaluated in this study are bigger than those of other researchers mentioned earlier. This is due to differences in assuming the applied region for calculating seepage force, and the lack of consideration of a lining and drainage material. 3. Theoretical solution of ground reaction curve with consideration of seepage forces 3.1. Theoretical solutions for stress It is assumed that a soil-mass behaves as an isotropic, homogeneous and permeable medium. Also, an elasto-plastic model based on a linear Mohr–Coulomb yield criterion is adopted in this study.
r01 ¼ kr03 þ ðk 1Þa
ð4Þ
Here r indicates the major principal stress, r is the minor principal stress, k ¼ tan2 45 þ /2 , a ¼ tanc /, k and a are the Mohr–Coulomb constants, c is the cohesion, and u is the friction angle. Fig. 7 shows a circular opening of radius r0 with k0 = 1 in an infinite soil-mass subject to a hydrostatic in situ stress, r00 . Considering all the stresses on an infinitesimal element abcd of unit thickness during excavation of a circular tunnel in Fig. 8, the equilibrium of radial forces with respect to r and h can be expressed as follows: 0 1
0 3
@ r0r @ r0 @h @r ðr þ @rÞ@h r0r r@h r0h þ h @h @r sin 2 @r @h @h @ rrh @h @h 0 rh @r sin þ rrh þ @h @r cos rrh @r cos 2 2 2 dh
r0r þ
þ F r r@r@h ¼ 0
ð5Þ
If oh is too small, sin (oh/2) may be replaced with oh/2, and cos (oh/2) by one. Additional simplification is achieved by dropping terms containing higher order infinitesimals. A similar analysis may be performed for the tangential direction. When both equilibrium equations are divided by roroh, the result is:
D=5m
@ r0r 1 @ rrh r0r r0h þ þ þ Fr ¼ 0 r @h @r r 0 0 0 1 @ rh @ rrh 2rrh þ þ þ Fh ¼ 0 r @h @r r
Fig. 4. The circular tunnel with various conditions.
ð7Þ
If the tunnel is excavated under the groundwater table, then it acts as a drain. The body force is the seepage stress, as illustrated in Fig. 8.
Table 1 The region for calculating seepage forces. The region for calculating seepage forces (m)
H/D = 2.5 H/D = 5.0 H/D = 7.5 H/D = 10
ð6Þ
C/D = 2.5
C/D = 5.0
C/D = 7.5
C/D = 10
12.5 12.5 12.5 12.5
12.5 25.0 25.0 25.0
12.5 25.0 37.5 37.5
12.5 25.0 37.5 50.0
F r ¼ ir cw
ð8Þ
F h ¼ ih cw
ð9Þ
In this state, ir and ih are the hydraulic gradient in r and h directions respectively, and cw is the unit weight of groundwater. Therefore Eqs. (6) and (7) can be rewritten as follows:
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Table 2 Seepage pressure acting on the tunnel. C/D = 2.5
H/D = 2.5 H/D = 5.0 H/D = 7.5 H/D = 10
C/D = 5.0
C/D = 7.5
C/D = 10
Seepage pressure (kN/m2)
Seepage pressure ratio (%)
Seepage pressure (kN/m2)
Seepage pressure ratio (%)
Seepage pressure (kN/m2)
Seepage pressure ratio (%)
Seepage pressure (kN/m2)
Seepage pressure ratio (%)
117.6 235.2 352.8 477.3
95.0 96.0 96.0 97.4
105.8 235.2 352.8 473.3
86.4 96.0 96.0 96.5
105.8 224.4 351.8 468.4
86.4 91.6 95.7 95.6
107.8 225.4 339.1 469.4
88.0 92.0 92.3 95.8
σ 0'
600
Elastic
Seepage pressure (kN/m2)
500
Plastic
400 i
σ
300
' 0
re
σ 0'
ro
200
100
0 0
2.5
5
7.5
10
σ 0'
12.5
Groundwater level ratio, H/D Fig. 7. Circular opening in an infinite medium. Fig. 5. Seepage pressure according to the variation of the groundwater level (C/ D = 10).
σ rθ + 100
σθ +
∂σ rθ ∂θ ∂θ
∂σ θ ∂θ ∂θ
Fθ
80
a θ
r∂
r Seepage pressure ratio (%)
σr +
d
60
σr
σ rθ
Fr
b
∂σ r ∂r ∂r
σ rθ + ∂r
c σ rθ
∂σ rθ ∂r ∂r
σθ
∂θ
40
20
O Tunnel
0 0
2.5
5
7.5
10
12.5
Fig. 8. Body forces under the groundwater table.
Groundwater level ratio, H/D
ð10Þ
If the stress distribution is symmetrical with respect to the axis O in Fig. 8, then the stress components are not varied with angular orientation, h, and therefore they are functions of the radial distance r only. Accordingly, Eq. (10) reduces to the single equation of equilibrium as follows:
ð11Þ
dr0r r0r r0h þ þ ir cw ¼ 0 dr r
Fig. 6. Seepage pressure ratio according to the variation of the groundwater level (C/D = 10).
@ r0r 1 @ r0rh r0r r0h þ þ þ ir cw ¼ 0 r @h @r r 1 @ r0h @ r0rh 2r0rh þ þ þ ih cw ¼ 0 r @h @r r
ð12Þ
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For the plastic region, Eq. (4) can be modified as follows: 0 h
0 r
r ¼ kr r þ ðkr 1Þar
ð13Þ
where kr ¼ tan 45 þ /2r , ar ¼ tancr/r , kr and ar are the Mohr–Coulomb constants, cr is the cohesion, and ur is the friction angle in plastic region. By substituting Eq. (13) into Eq. (12), Eq. (12) can be given as follows:
C r
A 4
B 2
r0rei ¼ 2 ½2 logðrÞ 1
1 2v 1 IðrÞ JðrÞ 2ðv 1Þ 2ð1 v Þ ð19Þ
2
dr0r 1 ð1 kr Þr0r þ ð1 kr Þar þ ir cw ¼ 0 þ r dr
ð14Þ
The above partial differential equation can be solved by using the boundary conditions r0r ¼ pi at r = r0. Then, the radial and circumferential effective stresses in the plastic region are as follows (Shin et al., submitted for publication):
r0rp ¼
r
0 hp
r 1kr 0 ðpi þ ar Þ ar r
Z r Z r0 cw 1kr 1kr 1k n i n i r ðnÞdn r ðnÞdn r r R0 R0
r 1kr 0 ¼ kr ðpi þ ar Þ ar kr r
Z r Z r0 cw 1kr 1kr 1k n i n i r ðnÞdn r ðnÞdn r r R0 R0
r 0 2 r 0 r 0 2 0 ¼ r0 þ r0 r
C A B 1 2v 1 ½2 logðrÞ þ 1 IðrÞ þ JðrÞ r2 4 2 2ðv 1Þ 2ð1 v Þ
ð20Þ
Rr Rr where IðrÞ ¼ cw R0 ir ðnÞdn, JðrÞ ¼ crw2 R0 n2 ir ðnÞdn, C, A and B are constants defined by the boundary conditions. By using the boundary conditions r0rei ¼ 0 at r = r0, r0rei ¼ 0 at r = R0, and r0hei ¼ 0 at r = R0, Eqs. (19) and (20) become as follows (i.e. R0: the distance of region for calculating seepage force):
Z
r0rei ¼ 1
þ
r 20
þ 1
ð15Þ
r 20
2 log R0 2 log r 0 þ1 R20
1 Z 1 ½2 logðrÞ 1 r 2 12 þ 2 log R0 22 log r0 þ1 R20 r R 0
Z
0
2 log R0 R20
log r0 þ1 þ 2 log R0 2 R2 0
1 2v 1 IðrÞ JðrÞ 2ðv 1Þ 2ð1 v Þ ð21Þ
ð16Þ
In this equation, pi is all the support pressure developed by in situ stress and seepage. Subscript rp and hp indicate the radial and tangential effective stresses in the plastic region respectively. In order to estimate the effective stress in elastic region, the superposition concept is used. As shown in Fig. 9, the effective stress considering the seepage force can be assumed as a combination of the solution of equilibrium equation in dry condition and the effective stress only considering seepage. The Kirsch solutions are applied to solve the effective stresses in the elastic region under dry condition (Timoshenko and Goodier, 1969). At the boundary conditions r = r0 and r = 1, substituting r0r ¼ pi and r0r ¼ r00 , then the effective stresses in elastic zone under a dry condition can be obtained as following Eqs. (17) and (18). Subscript re and he represent the radial and circumferential effective stress in elastic zone, and d indicates the dry condition, where r00 indicates in situ effective stress.
r0hei ¼
r0reðdÞ ¼ r00 r00
ð17Þ
r0heðdÞ
ð18Þ
r0hei ¼ 1 r 20
Z 1 Z 1 1 2 log R 2 log r þ1 2 ½2 logðrÞ log r 0 þ1 r 2 0 0 R þ 2 log R0 2 þ 2 2 2 0 r R R 0
þ 1 þ 1 r 20
0
2 log R0
log r 0 þ1 þ 2 log R0 2 R2 0
R20
1 IðrÞ 2ðv 1Þ
2v 1 þ JðrÞ 2ð1 v Þ
ð22Þ
Here
Z¼
1 c 2ðv 1Þ w
Z
r0
ir ðnÞdn þ
R0
2v 1 cw 2ð1 v Þ r 20
Z
r0
n2 ir ðnÞdn;
R0
and subscript i represents the term related to seepage. Consequently, the radial and tangential effective stresses with consideration into seepage forces in elastic region can be obtained by superposition of Eqs. (17) and (18) and Eqs. (21) and (22) as follows:
r0re ¼ 1 r 20
Z þ
þ 1 r 20
Stern (1965) suggests effective stresses in elastic region with consideration of seepage force as follows:
0
Z
2 log R0 2 log r 0 þ1 R20
1 Z 1 ½2 logðrÞ 1 r 2 12 þ 2 log R0 22 log r0 þ1 R20 r R
Z
0
2 log R0
log r 0 þ1 þ 2 log R0 2 R2 0
r 2 0 þ r00 ðr00 pi Þ r
Fig. 9. Concept of superposition in elastic region.
R20
0
1 2v 1 IðrÞ JðrÞ 2ðv 1Þ 2ð1 v Þ ð23Þ
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Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324
r0he ¼ 1 r 20
Z 1 Z 1 1 2 log R 2 log r þ1 2 ½2 logðrÞ þ 1 log r 0 þ1 r 2 0 0 þ 2 log R0 2 þ R 2 2 2 0 r R R 0
þ 1 r 20
0
Z þ
þ r00 þ
r00 pi
0
2 log R0
2 log R0 2 log r 0 þ1 R20
Eqs. (29)–(32) lead to the following differential equation as follows:
R20
1 2v 1 IðrÞ þ JðrÞ 2ðv 1Þ 2ð1 v Þ ð24Þ
r
1 0 1k 2r0 þ a 1þk 1þk 1 1 A½logðr e Þ B Iðr e Þ þ 1þk ðv 1Þ
A ¼ 1 r 20
4
ð25Þ
log r 0 þ1 2 R0 þ 2 log R0 2 R2
B ¼ 1
0
r 20
4 log R0
Z log r 0 þ1 þ 2 log R0 2 R2 0
ð26Þ
ð27Þ ð28Þ ð29Þ ð30Þ
The plastic strain can be represented by using the plastic flow rule. When the volume expansion effect is important in the plastic strain, generally the non-associated flow rule is valid and, otherwise, the associated flow rule is valid. The plastic potential function, Q in case of using non-associated flow rule, is as follows:
ð31Þ
w where kw ¼ 1þsin , the parameter w is the dilation angle. 1sin w The plastic parts of radial and circumferential strains can be related as follows (Wang, 1994):
epr ¼ kw eph
r kw f ðrÞdr þ urðr¼re Þ
r kw e r
ð32Þ
ð36Þ
1 D ð1 2tÞC þ 2 2G r
1 D ¼ ð1 2tÞC 2 2G r
eer ¼
ð37Þ
eeh
ð38Þ
r0rðr¼re Þ r00 r2e ðpi r00 Þr20
The radial displacement for a circular tunnel can be worked out based on the elasto-plastic theory. The strains in the plastic region are composed of elastic and plastic strain, and are expressed as Eqs. (27) and (28). The superscripts e and p represent the elastic and plastic parts, respectively. By considering compressive strains and radially inward displacements to be positive, the relationship between strain and displacement at any point in the soil-mass can be written as follows:
qffiffiffiffiffi kw ¼ 0
ð35Þ
In order to evaluate the integral in the above equation, expressions for eer and eeh can be obtained by following equation (Brady and Brown, 1993):
pi r0rðr¼r
eÞ r 2e r 20
r 20 r 2e
Here C ¼ , D¼ , and t is the r 2e r 20 Poisson’s ratio of the soil-mass. By substituting Eqs. (37) and (38) into Eq. (34), and then substituting the resulting expression for f(r) into Eq. (36), the integration of Eq. (36) can be calculated mathematically. The expression for the radial displacement in the plastic region can be obtained as follows (Sharan, 2003):
ur ¼
Q ¼ f ðrr ; rh Þ ¼ rh kw rr 2c
r
3.2. Theoretical solutions for displacement
dur er ¼ dr u eh ¼ r r
Z
re
er ¼ eer þ epr eh ¼ eeh þ eph
re ðrv o rrðr¼re Þ Þ 2G
where G is the shear modulus of the soil-mass. Eqs. (33)–(35) lead to the following expressions for the radial displacement:
R20
1 1 0 1k re ¼ r0 2r 0 þ a þ ar pi þ ar 1 þ k 1þk Z re 1 1 þ cw ir ðnÞdn A½logðr ec Þ B 1 þ k ðv 1Þ R0
Z r e k 11 Z r0 r cw 1kr 1kr þ 1k n i n i r ðnÞdn r ðnÞdn r re R0 R0
ð34Þ
Eq. (33) can be solved by using the following boundary condition for the radial displacement, urðr¼re Þ at the elasto-plastic interface (Brady and Brown, 1993).
ur ¼ r kw
Finally, at the interface between the plastic and elastic regions, the radial stress calculated in the plastic region must be identical to that in the elastic region. Consequently, Eq. (15) should be equal to Eq. (25) since the radial stress should be continuous over the boundary. The following Eq. (26) of the radius of the plastic zone, re can be derived as follows:
eer þ kw eeh ¼ f ðrÞ
urðr¼re Þ ¼
where
Z
ð33Þ
where
r 0 2
The following Eq. (25) is derived from Eqs. (23) and (24) and Mohr–Coulomb yield criterion at the stress state in the elastic region.
r0re ¼
dur ur þ kw ¼ f ðrÞ dr r
i 1 kw h k þ1 k 1 Cð1 2tÞ rew r kw þ1 D r ew r kw 1 r 2G r kw e þ urðr¼re Þ r
ð39Þ
The radial displacement urðr¼r0 Þ at the opening surface r = r0 is given by Eq. (40).
urðr¼r0 Þ ¼
i 1 kw h k þ1 k þ1 k 1 k 1 r 0 Cð1 2tÞ r ew r 0w D rew r0w 2G kw re ð40Þ þ urðr¼re Þ r0
The ground reaction curve is estimated through theoretical solution for the cases in which the cover depth of tunnel and the water height are 10 times diameter of tunnel in Fig. 4. In this paper, nondilating soil-mass is utilized (W = 0). As shown in Fig. 10, the ground reaction curve with consideration of seepage force is bigger than the ground reaction curve in dry condition. This is due to additional seepage forces to the initial effective overburden pressure. 3.3. Elastic stress induced by seepage forces In this study, the superposition concept is used to determine the effective stress considering seepage. In order words, the effective stress including seepage in elastic region is a combination of the effective stress in dry condition and the effective stress only considering seepage. The theoretical solution in elastic region only by seepage is compared with studies of other researchers and estimated. Muir Wood (1975) states that the solution of elastic stress by seepage is as follows:
Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324
321
where ho is the water depth from the groundwater table to tunnel
100
2
o . axis level, and T ¼ 1 þ 4h r2
On the other hand, Bobet (2003) proposes the stress solution by using a far field pore water pressure, uf for the elastic ground mass.
Dry condition
80
Seepage condition
0 r
r
60
pi(%) 40
20
r0h ¼
0
0.1
0.2
0.3
0.4
0.5
Fig. 10. The ground reaction curve (C/D = 10, H/D = 10).
qðr 2o ln r o r 2 ln rÞ 2kð1 mÞr2
q r 2 ln r o þ r 2 ln r þ r2 ¼ 1 o k 2ð1 mÞr 2
rr ¼
ð41Þ
rh
ð42Þ
In this case, ho is the water depth from the groundwater table to tunnel axis level, and q is flow rate into the tunnel per unit length of tunnel. Muir Wood (1975) derives theoretical solutions from the flux of flow into a tunnel. He insists that the stresses by water might add to ground load stresses to establish the overall condition of loading supported by the tunnel. Also, a lining with permeability relatively lower than that of the ground reduces the flow per unit length in Eqs. (41) and (42), thus leading to the reduction of ground loading, while the fraction of water load carried directly on the lining compensates for it. His solution contains a mistake in the boundary conditions. Curtis et al. (1976) modified that by changing boundary condition as follows:
rr ¼ ln R0 R20 r 20
r 20 R20
r 20
ln
! R0 q 2kð1 mÞ r0
r20 R20 q ln Rr00 2kð1 mÞ r2
q lnðrÞ 2kð1 mÞ
ð43Þ
! R0 q ¼ ln R0 ln 2kð1 mÞ r0 r20 R20 R20 r 20 q ln Rr00 2kð1 mÞ r 20 R20 q lnðrÞ q q þ 2kð1 mÞ 2kð1 mÞ k r2 r 20
ð44Þ
9 2 ln r þ ½ð1 2mÞT 2ð1 mÞ ln ðr2 r2o Þ> = 2 1 ro ro T o 1 þ 2 > 2ð1 mÞ > r ln T ; : 8 > <r 2
ð45Þ
r0h ¼ cw ho
8 <
1 r 2o þ1 2 2ð1 mÞ : r
h
i 9 2 ðr 2 r 2 Þ o þ 2m ln r2 T o = 2 ln rro ð1 2mÞ 1 þ 4h r2 o
ln T
ln Rro þ 1 2m
ð48Þ
As discussed above, Fig. 11 shows the elastic stress by the seepage forces due to excavation of tunnel below the groundwater table. The seepage stress, r0ei in the diagram is normalized with the hydrostatic pressure at the tunnel crown, Dpw and the radial distance is normalized with tunnel diameter, D. The theoretical solution by Muir Wood (1975) is illustrated in Fig. 11a with the correction by Curtis et al. (1976) shown in Fig. 11b. Fig. 11b shows the theoretical solution by Stern (1965), Fernandez and Alvarez (1994), and Curtis et al. (1976) together, and compares each to the others. Each of these solutions is very similar at the tunnel perimeter, but they have a tendency to increase in difference with distance from the tunnel due to boundary conditions. This is because Fernandez and Alvarez (1994) and Curtis et al. (1976) used the infinite boundary condition, r = 1, r0h ¼ 0 and Stern (1965) used the finite boundary, r = R0, r0h ¼ 0. However, these differences can be negligible in magnitude. Therefore, the usage of finite boundary is reasonable. Fig. 11c shows the theoretical solution by Bobet (2003), which proves to be different from others’ results, because the far field pore water pressure is considered as external load. According to the summary above, it is reasonable to apply the theoretical solution by Stern (1965) to estimate the seepage stress in elastic region. 4. Ground reaction curve considering seepage forces
Fernandez and Alvarez (1994) suggest the solution of elastic stress due to seepage under steady-state flow as follows:
r0r ¼ cw ho
ð47Þ
" # 2 uf Ro þ1 þ r 2ð1 mÞ
ln Rroo
r 2 uf o 1 ð1 2mÞ 2ð1 mÞ r
0.6
ur(m)
rh
1 2m r 2o uf 2 2ð1 mÞ Ro r 2o
0
þ
" # 2 ln Rro uf 1 2m r 2o Ro ¼ uf 1 þ 2 2ð1 mÞ Ro r 2o r 2ð1 mÞ ln Rr o o
r 2 uf o 1 þ ð1 2mÞ 2ð1 mÞ r
; ð46Þ
To validate the theoretical solution considering seepage forces, the ground reaction curves achieved by the numerical analysis and theoretical solutions are compared with each other and estimated. To verify the application of numerical analysis modeling used in this study, the ground reaction curves of the theoretical solution, which was suggested by Sharan (2003), and computed by numerical analysis in dry condition are compared. Based on this verification, the theoretical solutions of ground reaction curve considering seepage forces are evaluated and the applicability of theoretical solutions induced in this study is suggested. 4.1. General conditions for numerical analysis The program used in numerical analysis is PENTAGON 2D (Emerald Soft Consulting Co., 1998). It is a finite element analysis program for solving two-dimensional continuum problems and seepage problem. Both of the associated flow rule and non-associated flow rule are adopted when plastic strain is present after the material yields. The soil-mass is assumed to be governed by a
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Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324 Table 4 Cases for analysis under dry condition (H/D = 0).
a
C/D
Dry condition
′ σ rei
5.0
10.0
15.0
Case 1
Case 2
Case 3
Δpw
σ θ′ei
σ ei′
Table 5 Cases for analysis when considering seepage forces.
Δpw
Δpw
H/D
C/D 5.0
2.5 5.0 7.5 10.0
Δpw
7.5 21 22 23 24
Case Case Case Case
10.0 31 32 33 34
Case Case Case Case
41 42 43 44
the tunnel depth, C and groundwater table, H as shown in Tables 4 and 5. The boundary conditions for seepage analysis are shown in Fig. 12. To carry out the seepage analysis, a total head is applied at the outer boundary before the tunnel excavation, and only the elevation head is applied with its pressure head being zero at the cross-section of the tunnel after excavation. To get the ground reaction curve by the numerical analysis, the internal pressure acting on the tunnel decreases with the ten steps by ten percent, and the radial displacements of the tunnel are found at each step.
b
σ ei′
Case Case Case Case
σ θ′ ei Δpw
4.2. Procedure for evaluating the ground reaction curve by theoretical solution
′ σ rei Δpw
The ground reaction curve by theoretical solution can be obtained from the equations derived before, and the general procedure is as follows: first, calculate hydraulic gradient distribution using numerical analysis in the case of seepage condition (seepage analysis is not needed under dry condition); assume the value of internal pressure, pi; calculate the outer boundary of plastic region, re and radial displacement, ur at the cross-section of tunnel by using Eqs. (26) and (40), respectively; and complete the ground reaction curve, which represents the relationship between internal pressure and radial displacement by repeating the above process with newly assumed values of internal pressure.
c ′ σ rei Δpw σθe ' ′ σ θ ei Δpw Δpw
σ re '
Δpw
σ ei′ Δpw
4.3. Comparison between analytical solution and numerical analysis
Fig. 11. The theoretical solution of seepage stress in elastic region. (a) The theoretical solution of seepage stress in elastic region by Muir Wood (1975). (b) The theoretical solutions of seepage stress in elastic region. (c) The theoretical solution of seepage stress in elastic region by Bobet (2003).
linear Mohr–Coulomb yield criterion. The properties of the ground materials used in the analysis are shown in Table 3. A circular tunnel with a diameter of 5.0 m, as shown in Fig. 4, is analyzed, and the ground reaction curve is calculated by changing
To verify the feasibility of the numerical analysis, the ground reaction curves from the theoretical solution and numerical method under dry conditions (H/D = 0) suggested by Sharan (2003) are obtained and compared with each other. Fig. 13 shows the ground reaction curves at tunnel crown under dry conditions by both theoretical solution and numerical analysis in the case that the cover depths of the tunnel, C were the 5, 10, and 15 times the diameter of tunnel, D respectively. The ground reaction curves obtained by both methods are in close agreement. Therefore, the numerical analysis modeling carried out in this study is appropriate. To verify the theoretical solution considering seepage forces induced in this study, the ground reaction curves by the theoretical solution are compared with those by numerical analysis. For these cases, the cover depths of tunnel, C are the 5 times, 7.5 times and
Table 3 Properties of ground for analysis. Soil type
Submerged unit weight, csub (kN/m3)
Cohesion, c (kN/m2)
Internal friction angle, / (deg)
Dilation angle, w (deg)
Elastic modulus, E (kN/m2)
Poisson’s ratio
Weathered soil
4.9
9.81
35.0
0.0
49,000
0.30
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Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324
100
pi (%)
Theoretical solution-H/D=2.5
90
Theoretical solution-H/D=5.0
80
Theoretical solution-H/D=7.5
70
Theoretical solution-H/D=10.0 Numerical solution-H/D=2.5
60 50
Numerical solution-H/D=5.0
40
Numerical solution-H/D=7.5 Numerical solution-H/D=10.0
30 20 10 0 0
0.2
0.4
0.6
0.8
1.0
Radial displacement (m) Fig. 14. The ground reaction curves with consideration of seepage force (C/D = 5.0).
100 Fig. 12. Boundary conditions for numerical analysis.
100
Theoretical solution-C/D=5
90 80 70 60
pi (%)
pi (%)
Theoretical solution-H/D=5.0
80
Theoretical solution-H/D=7.5
70
Theoretical solution-H/D=10.0 Numerical solution-H/D=2.5
60
Theoretical solution-C/D=10
50
Numerical solution-H/D=5.0
Theoretical solution-C/D=15
40
Numerical solution-C/D=5
30
Numerical solution-H/D=7.5 Numerical solution-H/D=10.0
20
Numerical solution-C/D=10
50
Theoretical solution-H/D=2.5
90
10
Numerical solution-C/D=15
0
40
0
0.2
0.4
0.6
0.8
1.0
Radial displacement (m)
30 20
Fig. 15. The ground reaction curves with consideration of seepage (C/D = 7.5).
10
100
0 0
0.03
0.06
0.09
0.12
Radial displacement (m) Fig. 13. The ground reaction curves under dry condition according to the C/D variation.
10 times diameter of tunnel, D with the variation of groundwater table, H from the 2.5 times to 10 times diameter of tunnel. The ground reaction curves are drawn in Figs. 14–16 respectively. It is found that the results of the numerical method are almost similar with those of the theoretical solution. The theoretical solutions of ground reaction curve with consideration of seepage forces are verified with numerical analysis. However, the differences between the results of theoretical solution and numerical analysis can be bigger than those in the case of dry conditions in the same cases. The reason for this difference is due to the method in which seepage is dealt with in the theoretical solution with the consideration of seepage. As mentioned before, the groundwater would flow non-radially in most cases. This means that seepage considered in theoretical solution can be different from the actual flow. For the analysis by theoretical solution, the only radial component of hydraulic gradient distribution is calculated by the numerical analysis, and the seepage forces are then added by integration of the radial component of hydraulic gradient to the theoretical solution. This means that the whole seepage may not be reflected in the solution in some conditions. In case of the numerical method, however, after the hydraulic gradient is calculated by the total head loss at all meshes from groundwater table to tunnel wall, all the seepage forces are accumulated from each mesh acting on the tunnel, and are applied to the analysis. Therefore, this discrepancy between the two cases may account for the differences in the results,
pi (%)
Theoretical solution-H/D=2.5
90
Theoretical solution-H/D=5.0
80
Theoretical solution-H/D=7.5
70
Theoretical solution-H/D=10.0 Numerical solution-H/D=2.5
60 50
Numerical solution-H/D=5.0
40
Numerical solution-H/D=7.5 Numerical solution-H/D=10.0
30 20 10 0 0
0.2
0.4
0.6
0.8
1.0
Radial displacement (m) Fig. 16. The ground reaction curves with consideration of seepage (C/D = 10.0).
as shown in Fig. 14 (H/D = 10.0, C/D = 5.0) and Fig. 16 (H/D = 2.5, C/ D = 10.0). These show the bigger discrepancies when compared with other cases. The first case is twice higher groundwater table than cover depth (H/C = 2) and the second one is the case of four times lower groundwater table than cover depth (H/C = 0.25). The fact that there are big differences in the case of too high or too low water height relative to cover depth shows the limit of usage of the theoretical solution proposed in this study. 4.4. The applicability of the theoretical solution of ground reaction curves As discussed in the previous chapter, according to the comparison of the theoretical solution and numerical analysis, the theoretical solution of this study can be applied in the cases where the ratio of
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water height and cover depth, H/C is ranged between 0.25 and 2 (0.25 6 H/C 6 2). Moreover, the theoretical solution in this study is valid for radial flow. The ground water flow is essentially radial when the ratio of water height and diameter of tunnel, H/D is over 10, in cases where the water height equals to cover depth (Fernandez and Alvarez, 1994). These are applicable limits of the theoretical solution derived when considering seepage in this study. 5. Conclusions In this study, the effect of seepage force on tunnels is investigated using analytical and numerical methods. Theoretical solutions including seepage forces are proposed, and validated using numerical methods. It is found that the influence of seepage force on the ground reaction curve is significant. The conclusions reached and the observations provided contribute to a better understanding of the behavior of tunneling below the groundwater table. The results obtained from this study can be summarized as follows: (1) The range for calculating seepage forces is defined as the distance where the radial hydraulic gradient becomes to zero. This is because the distinct failure plane does not occur in tunnels different from earth retaining walls and tunnel faces. When the groundwater table is below ground surface, the seepage pressure ratios show the range of 85–90% of the hydrostatic pressure at a tunnel crown. When the groundwater table was higher than the ground surface, the seepage pressure ratios is over 95% of the hydrostatic pressure at tunnel crown. (2) It is shown that the flow of groundwater into tunnels results in significant increase in the radial displacements of the tunnel wall due to seepage force. (3) In comparison with dry conditions, the existence of flow into a tunnel causes seepage force and consequently has reduced the arching effect during excavation and large displacements. (4) The theoretical solution of ground reaction curve with consideration of seepage forces is verified with numerical analyses. Through the comparison of theoretical solution and numerical analysis, it is found that the theoretical solution in this study is applicable when the ratio of water height and cover depth, H/C ranges from 0.25 to 2, and when the ratio of water height and tunnel diameter, H/D is over 10 for an equal ratio of water height and cover depth (H/C = 1). Acknowledgement This paper was supported by the Underground Space Construction Technology Center under the Ministry of Construction and Transportation in Korea (Grant C04-01).
References Atkinson, J.H., Mair, R.J., 1983. Loads on leaking and watertight tunnel lining, sewers and buried pipes due to groundwater. Geotechnique 33 (3), 341–344. Bobet, A., 2003. Effect of pore water pressure on tunnel support during static and seismic loading. Tunnelling and Underground Space Technology 18, 377–393. Brady, B.H.G., Brown, E.T., 1993. Rock Mechanics for Underground Mining. Chapman and Hall, London. Brown, E.T., Bray, J.W., Ladanyi, B., Hoek, E., 1983. Ground response curve for rock tunnels. Journal of Geotechnical Engineering 109 (1), 15–39. Carranza-Torres, C., 2002. Dimensionless graphical representation of the exact elasto-plastic solution of a circular tunnel in a Mohr–Coulomb material subject to uniform far-field stresses. Rock Mechanics and Rock Engineering 36 (3), 237– 253. Carranza-Torres, C., Fairhurst, C., 1999. The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion. International Journal of Rock Mechanics and Mining Sciences 36, 777–809. Carranza-Torres, C., Fairhurst, C., 2000. Application of the convergence-confinement method of tunnel design to rock masses that satisfy the Hoek–Brown failure criterion. Tunnelling and Underground Space Technology 15 (2), 187–213. Carranza-Torres, C., Zhao, J., 2009. Exact and approximate solutions for displacements around circular openings in elastic–brittle–plastic Hoek– Brown rock. International Journal of Rock Mechanics and Mining Sciences 42, 542–549. Curtis, D.J., Mott, Hay, Anderson, 1976. The circular tunnel in elastic ground. Geotechnique 26 (1), 231–237. Emerald Soft Consulting Co., 1998. PENTAGON 2-D user manual. Seoul. Fernandez, G., 1994. Behavior of pressure tunnels and guidelines for liner design. Journal of Geotechnical Engineering 120 (10), 1768–1791. Fernandez, G., Alvarez, T.A., 1994. Seepage-induced effective stresses and water pressures around pressure tunnels. Journal of Geotechnical Engineering 120 (1), 108–128. Lee, I.M., Nam, S.W., 2001. The study of seepage forces acting on the tunnel lining and tunnel face in shallow tunnels. Tunnelling and Underground Space Technology 16, 31–40. Lee, S.W., Jung, J.W., Nam, S.W., Lee, I.M., 2007. The influence of seepage forces on ground reaction curve of circular opening. Tunnelling and Underground Space Technology 21, 28–38. Muir Wood, A.M., 1975. The circular tunnel in elastic ground. Geotechnique 25 (1), 115–127. Oreste, P.P., 2003. Analysis of structural interaction in tunnels using the convergence-confinement approach. Tunnelling and Underground Space Technology 18, 347–363. Schweiger, H.F., Pottler, R.K., Steiner, H., 1991. Effect of seepage forces on the shotcrete lining of a large undersea cavern. In: Beer, Booker, Carter (Eds.), Computer Method and Advances in Geomechanics, pp. 1503–1508. Sharan, S.K., 2003. Elastic–brittle–plastic analysis of circular openings in Hoek– Brown media. International Journal of Rock Mechanics and Mining Sciences 40, 817–824. Sharan, S.K., 2005. Exact and approximate solutions for displacements around circular openings in elastic–brittle–plastic Hoek–Brown rock. International Journal of Rock Mechanics and Mining Sciences 42, 542–549. Shin, J.H., Potts, D.M., Zdravkovic, L., 2005. The effect of pore-water pressure on NATM tunnel linings in decomposed granite soil. Canadian Geotechnical Journal 42, 1585–1599. Shin, J.H., Lee, I.M., Shin, Y.J., submitted for publication. Seepage-induced stress due to tunnelling under drainage condition. International Journal for Numerical and Analytical Methods in Geomechanics. Stern, M., 1965. Rotationally symmetric plane stress distribution. Zeitschrift fur Angewandte Mathematik und Mechanik 45 (6), 446–447. Stille, H., Holmberg, M., Nord, G., 1989. Support of weak rock with grouted bolts and shotcrete. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 26 (1), 99–113. Timoshenko, S.P., Goodier, J.N., 1969. Theory of Elasticity. McGraw-Hill, New York. Wang, Yarlong., 1994. Ground response of circular tunnels in poorly consolidated rock. Journal of Geotechnical Engineering 122 (9), 703–708.
Contents lists available at ScienceDirect
Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
The ground reaction curve of underwater tunnels considering seepage forces Young-Jin Shin a, Byoung-Min Kim b, Jong-Ho Shin c, In-Mo Lee d,* a
Civil Works Division, Samsung C&T, Seoul, Republic of Korea Underground Space Construction Technology Centre, Seoul, Republic of Korea c Department of Civil Engineering, Konkuk University, Seoul, Republic of Korea d Department of Civil Engineering, Korea University, Seoul, Republic of Korea b
a r t i c l e
i n f o
Article history: Received 17 November 2008 Received in revised form 28 December 2009 Accepted 6 January 2010 Available online 18 February 2010 Keywords: Tunnel Seepage forces Ground reaction curve Theoretical solution
a b s t r a c t When a tunnel is excavated below groundwater table, the groundwater flows into the excavated wall of tunnel and seepage forces are acting on the tunnel wall. Such seepage forces significantly affect the ground reaction curve which is defined as the relationship between internal pressure and radial displacement of tunnel wall. In this paper, seepage forces arising from the ground water flow into a tunnel were estimated quantitatively. Magnitude of seepage forces was estimated based on hydraulic gradient distribution around tunnel. Using these results, the theoretical solutions of ground reaction curve with consideration of seepage forces under steady-state flow were derived. A no-support condition was taken into account. The theoretical solution derived in this study was validated by numerical analysis. The changes in the ground reaction curve according to various ground and groundwater conditions were investigated. Based on the results, the application limit of theoretical solution was suggested. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction When a tunnel is excavated below groundwater table, the groundwater may flow into the tunnel, and consequently seepage forces will occur through the ground. This may seriously influence on the behavior of the tunnel. The ground behavior due to tunneling can be understood theoretically by the convergence-confinement method. The convergence-confinement method is based on the principle in which a tunnel is stabilized by controlling its displacements after installation of the support near the tunnel face. The convergence-confinement method consists of three elements: the longitudinal deformation profile, the ground reaction curve, and the support characteristic curve. The longitudinal deformation profile, based on the assumption of nosupport, shows the radial displacement of the tunnel cross-section in the longitudinal direction from the tunnel face. The support characteristic curve describes the increasing pressure that acts on the supports as the radial displacement increases. Lastly, the ground reaction curve shows the increasing trends of radial displacement as the internal pressure of the tunnel decreases. Tunneling under ground water table induces additional seepage stress (Shin et al., submitted for publication) and the seepage forces are likely to have a strong influence on the ground reaction curve.
* Corresponding author. Tel.: +82 2 3290 3314; fax: +82 2 928 7656. E-mail address: [email protected] (I.-M. Lee). 0886-7798/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.tust.2010.01.005
The researches on ground reaction curve by Brown et al. (1983), Stille et al. (1989), Wang (1994), Carranza-Torres and Fairhurst (1999, 2000), Carranza-Torres (2002), Sharan (2003), and Oreste (2003) were studied without considering the seepage forces; they were only for the dry condition. The studies about the effects of seepage forces on the tunnel face or the support system by Muir Wood (1975), Curtis et al. (1976), Atkinson and Mair (1983), Schweiger et al. (1991), Fernandez and Alvarez (1994), Fernandez (1994), Lee and Nam (2001), Bobet (2003), Shin et al. (2005) were conducted. An elastic–brittle–plastic strain analysis of displacements around circular openings in an isotropic Hoek–Brown rock was performed by Sharan (2005) and the effect of water pressure on the mechanical response of cylindrical lined tunnels in elastoplastic media was investigated by Carranza-Torres and Zhao (2009). A simplified analytical solution of the ground reaction curve was suggested by Lee et al. (2007); however, mathematical solutions of ground reaction curves influenced by seepage forces have not yet been suggested. In this study, based on these previous studies, seepage forces due to tunneling under groundwater table were estimated quantitatively, and the theoretical solution of ground reaction curve considering seepage forces due to groundwater flow under steady-state flow was derived. The studies were performed for a no-support. The validity of the derived theoretical solution was examined by numerical analysis. The changes in the ground reaction curve according to various ground and ground water table conditions were investigated. Based on the results, the application limit of theoretical solution was suggested.
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2. Calculation of seepage forces 2.1. The range for calculating seepage forces
flow line
When groundwater flows into a tunnel below the groundwater table, the stress of the soil-mass would be the state of ‘‘the submerged weight plus seepage force”. In this case, seepage forces are considered one of the most influencing factors, and the calculation of seepage forces is crucial in the tunnel behavior. Seepage forces per unit volume can be formulated as shown in the following equation:
S:F=v ol: ¼
izcw A ¼ icw zA
iθ = i × sin α
ir
α
iθ
i
ð1Þ
where the hydraulic gradient, i is defined as the total head loss, Dh divided by the distance between any two points, cw is unit weight of water, A is area, and z is depth. Thus, seepage forces can be easily calculated if the total head loss, Dh is obtained at any two points. In this case, seepage forces can be calculated by obtaining the total head loss from the potential failure plane to the drainage region, and the direction of the seepage force is equal to that of flow line. In a tunnel, although seepage forces develop, the plastic region may or may not be encountered. Thus it is difficult to calculate seepage forces influencing a tunnel behavior quantitatively by using the method that was used in calculating seepage forces acting on the earth retaining structure and tunnel face. Therefore, the region for calculating seepage forces should be justified in advance to evaluate seepage forces in a tunnel. While tunneling under steady-state water flow, the groundwater flows into tunnel and the flow net as shown in Fig. 1 develops. The hydraulic gradients touching all the flow lines can be divided into a radial component of hydraulic gradient, ir and a tangential component of hydraulic gradient, ih with the angle, a as shown in Fig. 2. The radial component of hydraulic gradient, ir is used to reflect the seepage effect of tunneling in this study. This is because the radial component of the groundwater flow affects most of the displacement of tunnel wall, and the seepage forces can be estimated in both of the radial and non-radial flows by using radial component hydraulic gradient. All flows induced by tunneling under the groundwater table affect seepage forces. Therefore, the distance where the radial component of hydraulic gradient becomes zero is defined as the range for calculating seepage forces under steady-state groundwater flow, as shown in Fig. 3. Lee et al. (2007) define the range for calculating seepage forces until the critical hydraulic gradient, ir = 1 while this study extends to ir = 0.
i
ir = i × cos α
i
Fig. 2. The radial and tangential component of hydraulic gradient.
i α
i
α
r
ir = i × cos α
ir
Critical hydraulic gradient
ir = 1
Region for calculating seepage force
ir = 0
i
r
Fig. 3. The range for calculating seepage forces.
2.2. Calculation of seepage forces
Fig. 1. Flow net while tunneling under steady-state water flow.
If the lining is fully drained, ground water will seep toward the tunnel, and the seepage stress, r0i will increase due to the drag of water flowing past the soil grains. It acts in the direction of flow. Atkinson and Mair (1983) propose Eq. (2) as follows:
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dr0i ¼ cw i dS
ð2Þ
For Atkinson’s equation, r0i indicates seepage forces per volume, i is hydraulic gradient, and S is distance. If groundwater level is not varied and only flows into the radial direction, then Eq. (2) can be rewritten as following:
r0i ¼ cw
Z
ir dr
ð3Þ
where ir is the radial component of hydraulic gradient, and r is the radial distance. For the cases in which the cover depth of tunnel, C are 2.5 times, 5 times, 7.5 times and 10 times diameter of tunnel, parametric studies to investigate seepage forces are conducted with the variation of ground water height, H from 2.5 times to 10 times diameter of tunnel in a tunnel as shown like Fig. 4. The tunnel diameter remains 5 m in these parametric studies. The region for calculating seepage forces described above is shown in Table 1, the seepage forces obtained by using the above method are summarized in Table 2. The seepage pressure ratio in Table 2 is determined by generalizing seepage forces with hydrostatic pressure at the tunnel crown in a hydrostatic condition. Fig. 5 illustrates variation of seepage pressure as a function of ground water level with constant cover depth. As shown in Fig. 5, seepage pressure increases as the groundwater level increases. Fig. 6 shows the seepage pressure ratio. If the groundwater level is below the ground surface, though the seepage force acting on the tunnel is different depending on the groundwater table, the seepage pressure ratios show the range of 85–90% of the hydrostatic pressure at the tunnel crown. If the groundwater level is higher than the ground surface, the seepage pressure ratios are over 95%. Through several conducted studies, there have been different proposed ranges regarding quantification of the seepage force.
Schweiger et al. (1991) state that the magnitude of seepage force is 50–80% of the hydrostatic pressure. Lee and Nam (2001) expect that the magnitude of seepage force is 30–50% of hydrostatic pressure and Lee et al. (2007) propose it to be 50–80% of hydrostatic pressure. The differences are originated by different analytical methods. Schweiger et al. (1991) consider the deformation of ground and only first lining in calculating seepage forces. Thus, the estimated seepage force was less than the hydrostatic pressure. On the other hand, Lee and Nam (2001) consider secondary lining and a cushion effect created by drainage material in addition to the first lining. This is why the seepage force found by Lee and Nam is smaller than that found by Schweiger. Lee et al. (2007) propose a simple method for calculating seepage force by introducing the concept ‘‘critical hydraulic gradient”. The seepage forces evaluated in this study are bigger than those of other researchers mentioned earlier. This is due to differences in assuming the applied region for calculating seepage force, and the lack of consideration of a lining and drainage material. 3. Theoretical solution of ground reaction curve with consideration of seepage forces 3.1. Theoretical solutions for stress It is assumed that a soil-mass behaves as an isotropic, homogeneous and permeable medium. Also, an elasto-plastic model based on a linear Mohr–Coulomb yield criterion is adopted in this study.
r01 ¼ kr03 þ ðk 1Þa
ð4Þ
Here r indicates the major principal stress, r is the minor principal stress, k ¼ tan2 45 þ /2 , a ¼ tanc /, k and a are the Mohr–Coulomb constants, c is the cohesion, and u is the friction angle. Fig. 7 shows a circular opening of radius r0 with k0 = 1 in an infinite soil-mass subject to a hydrostatic in situ stress, r00 . Considering all the stresses on an infinitesimal element abcd of unit thickness during excavation of a circular tunnel in Fig. 8, the equilibrium of radial forces with respect to r and h can be expressed as follows: 0 1
0 3
@ r0r @ r0 @h @r ðr þ @rÞ@h r0r r@h r0h þ h @h @r sin 2 @r @h @h @ rrh @h @h 0 rh @r sin þ rrh þ @h @r cos rrh @r cos 2 2 2 dh
r0r þ
þ F r r@r@h ¼ 0
ð5Þ
If oh is too small, sin (oh/2) may be replaced with oh/2, and cos (oh/2) by one. Additional simplification is achieved by dropping terms containing higher order infinitesimals. A similar analysis may be performed for the tangential direction. When both equilibrium equations are divided by roroh, the result is:
D=5m
@ r0r 1 @ rrh r0r r0h þ þ þ Fr ¼ 0 r @h @r r 0 0 0 1 @ rh @ rrh 2rrh þ þ þ Fh ¼ 0 r @h @r r
Fig. 4. The circular tunnel with various conditions.
ð7Þ
If the tunnel is excavated under the groundwater table, then it acts as a drain. The body force is the seepage stress, as illustrated in Fig. 8.
Table 1 The region for calculating seepage forces. The region for calculating seepage forces (m)
H/D = 2.5 H/D = 5.0 H/D = 7.5 H/D = 10
ð6Þ
C/D = 2.5
C/D = 5.0
C/D = 7.5
C/D = 10
12.5 12.5 12.5 12.5
12.5 25.0 25.0 25.0
12.5 25.0 37.5 37.5
12.5 25.0 37.5 50.0
F r ¼ ir cw
ð8Þ
F h ¼ ih cw
ð9Þ
In this state, ir and ih are the hydraulic gradient in r and h directions respectively, and cw is the unit weight of groundwater. Therefore Eqs. (6) and (7) can be rewritten as follows:
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Table 2 Seepage pressure acting on the tunnel. C/D = 2.5
H/D = 2.5 H/D = 5.0 H/D = 7.5 H/D = 10
C/D = 5.0
C/D = 7.5
C/D = 10
Seepage pressure (kN/m2)
Seepage pressure ratio (%)
Seepage pressure (kN/m2)
Seepage pressure ratio (%)
Seepage pressure (kN/m2)
Seepage pressure ratio (%)
Seepage pressure (kN/m2)
Seepage pressure ratio (%)
117.6 235.2 352.8 477.3
95.0 96.0 96.0 97.4
105.8 235.2 352.8 473.3
86.4 96.0 96.0 96.5
105.8 224.4 351.8 468.4
86.4 91.6 95.7 95.6
107.8 225.4 339.1 469.4
88.0 92.0 92.3 95.8
σ 0'
600
Elastic
Seepage pressure (kN/m2)
500
Plastic
400 i
σ
300
' 0
re
σ 0'
ro
200
100
0 0
2.5
5
7.5
10
σ 0'
12.5
Groundwater level ratio, H/D Fig. 7. Circular opening in an infinite medium. Fig. 5. Seepage pressure according to the variation of the groundwater level (C/ D = 10).
σ rθ + 100
σθ +
∂σ rθ ∂θ ∂θ
∂σ θ ∂θ ∂θ
Fθ
80
a θ
r∂
r Seepage pressure ratio (%)
σr +
d
60
σr
σ rθ
Fr
b
∂σ r ∂r ∂r
σ rθ + ∂r
c σ rθ
∂σ rθ ∂r ∂r
σθ
∂θ
40
20
O Tunnel
0 0
2.5
5
7.5
10
12.5
Fig. 8. Body forces under the groundwater table.
Groundwater level ratio, H/D
ð10Þ
If the stress distribution is symmetrical with respect to the axis O in Fig. 8, then the stress components are not varied with angular orientation, h, and therefore they are functions of the radial distance r only. Accordingly, Eq. (10) reduces to the single equation of equilibrium as follows:
ð11Þ
dr0r r0r r0h þ þ ir cw ¼ 0 dr r
Fig. 6. Seepage pressure ratio according to the variation of the groundwater level (C/D = 10).
@ r0r 1 @ r0rh r0r r0h þ þ þ ir cw ¼ 0 r @h @r r 1 @ r0h @ r0rh 2r0rh þ þ þ ih cw ¼ 0 r @h @r r
ð12Þ
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Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324
For the plastic region, Eq. (4) can be modified as follows: 0 h
0 r
r ¼ kr r þ ðkr 1Þar
ð13Þ
where kr ¼ tan 45 þ /2r , ar ¼ tancr/r , kr and ar are the Mohr–Coulomb constants, cr is the cohesion, and ur is the friction angle in plastic region. By substituting Eq. (13) into Eq. (12), Eq. (12) can be given as follows:
C r
A 4
B 2
r0rei ¼ 2 ½2 logðrÞ 1
1 2v 1 IðrÞ JðrÞ 2ðv 1Þ 2ð1 v Þ ð19Þ
2
dr0r 1 ð1 kr Þr0r þ ð1 kr Þar þ ir cw ¼ 0 þ r dr
ð14Þ
The above partial differential equation can be solved by using the boundary conditions r0r ¼ pi at r = r0. Then, the radial and circumferential effective stresses in the plastic region are as follows (Shin et al., submitted for publication):
r0rp ¼
r
0 hp
r 1kr 0 ðpi þ ar Þ ar r
Z r Z r0 cw 1kr 1kr 1k n i n i r ðnÞdn r ðnÞdn r r R0 R0
r 1kr 0 ¼ kr ðpi þ ar Þ ar kr r
Z r Z r0 cw 1kr 1kr 1k n i n i r ðnÞdn r ðnÞdn r r R0 R0
r 0 2 r 0 r 0 2 0 ¼ r0 þ r0 r
C A B 1 2v 1 ½2 logðrÞ þ 1 IðrÞ þ JðrÞ r2 4 2 2ðv 1Þ 2ð1 v Þ
ð20Þ
Rr Rr where IðrÞ ¼ cw R0 ir ðnÞdn, JðrÞ ¼ crw2 R0 n2 ir ðnÞdn, C, A and B are constants defined by the boundary conditions. By using the boundary conditions r0rei ¼ 0 at r = r0, r0rei ¼ 0 at r = R0, and r0hei ¼ 0 at r = R0, Eqs. (19) and (20) become as follows (i.e. R0: the distance of region for calculating seepage force):
Z
r0rei ¼ 1
þ
r 20
þ 1
ð15Þ
r 20
2 log R0 2 log r 0 þ1 R20
1 Z 1 ½2 logðrÞ 1 r 2 12 þ 2 log R0 22 log r0 þ1 R20 r R 0
Z
0
2 log R0 R20
log r0 þ1 þ 2 log R0 2 R2 0
1 2v 1 IðrÞ JðrÞ 2ðv 1Þ 2ð1 v Þ ð21Þ
ð16Þ
In this equation, pi is all the support pressure developed by in situ stress and seepage. Subscript rp and hp indicate the radial and tangential effective stresses in the plastic region respectively. In order to estimate the effective stress in elastic region, the superposition concept is used. As shown in Fig. 9, the effective stress considering the seepage force can be assumed as a combination of the solution of equilibrium equation in dry condition and the effective stress only considering seepage. The Kirsch solutions are applied to solve the effective stresses in the elastic region under dry condition (Timoshenko and Goodier, 1969). At the boundary conditions r = r0 and r = 1, substituting r0r ¼ pi and r0r ¼ r00 , then the effective stresses in elastic zone under a dry condition can be obtained as following Eqs. (17) and (18). Subscript re and he represent the radial and circumferential effective stress in elastic zone, and d indicates the dry condition, where r00 indicates in situ effective stress.
r0hei ¼
r0reðdÞ ¼ r00 r00
ð17Þ
r0heðdÞ
ð18Þ
r0hei ¼ 1 r 20
Z 1 Z 1 1 2 log R 2 log r þ1 2 ½2 logðrÞ log r 0 þ1 r 2 0 0 R þ 2 log R0 2 þ 2 2 2 0 r R R 0
þ 1 þ 1 r 20
0
2 log R0
log r 0 þ1 þ 2 log R0 2 R2 0
R20
1 IðrÞ 2ðv 1Þ
2v 1 þ JðrÞ 2ð1 v Þ
ð22Þ
Here
Z¼
1 c 2ðv 1Þ w
Z
r0
ir ðnÞdn þ
R0
2v 1 cw 2ð1 v Þ r 20
Z
r0
n2 ir ðnÞdn;
R0
and subscript i represents the term related to seepage. Consequently, the radial and tangential effective stresses with consideration into seepage forces in elastic region can be obtained by superposition of Eqs. (17) and (18) and Eqs. (21) and (22) as follows:
r0re ¼ 1 r 20
Z þ
þ 1 r 20
Stern (1965) suggests effective stresses in elastic region with consideration of seepage force as follows:
0
Z
2 log R0 2 log r 0 þ1 R20
1 Z 1 ½2 logðrÞ 1 r 2 12 þ 2 log R0 22 log r0 þ1 R20 r R
Z
0
2 log R0
log r 0 þ1 þ 2 log R0 2 R2 0
r 2 0 þ r00 ðr00 pi Þ r
Fig. 9. Concept of superposition in elastic region.
R20
0
1 2v 1 IðrÞ JðrÞ 2ðv 1Þ 2ð1 v Þ ð23Þ
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Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324
r0he ¼ 1 r 20
Z 1 Z 1 1 2 log R 2 log r þ1 2 ½2 logðrÞ þ 1 log r 0 þ1 r 2 0 0 þ 2 log R0 2 þ R 2 2 2 0 r R R 0
þ 1 r 20
0
Z þ
þ r00 þ
r00 pi
0
2 log R0
2 log R0 2 log r 0 þ1 R20
Eqs. (29)–(32) lead to the following differential equation as follows:
R20
1 2v 1 IðrÞ þ JðrÞ 2ðv 1Þ 2ð1 v Þ ð24Þ
r
1 0 1k 2r0 þ a 1þk 1þk 1 1 A½logðr e Þ B Iðr e Þ þ 1þk ðv 1Þ
A ¼ 1 r 20
4
ð25Þ
log r 0 þ1 2 R0 þ 2 log R0 2 R2
B ¼ 1
0
r 20
4 log R0
Z log r 0 þ1 þ 2 log R0 2 R2 0
ð26Þ
ð27Þ ð28Þ ð29Þ ð30Þ
The plastic strain can be represented by using the plastic flow rule. When the volume expansion effect is important in the plastic strain, generally the non-associated flow rule is valid and, otherwise, the associated flow rule is valid. The plastic potential function, Q in case of using non-associated flow rule, is as follows:
ð31Þ
w where kw ¼ 1þsin , the parameter w is the dilation angle. 1sin w The plastic parts of radial and circumferential strains can be related as follows (Wang, 1994):
epr ¼ kw eph
r kw f ðrÞdr þ urðr¼re Þ
r kw e r
ð32Þ
ð36Þ
1 D ð1 2tÞC þ 2 2G r
1 D ¼ ð1 2tÞC 2 2G r
eer ¼
ð37Þ
eeh
ð38Þ
r0rðr¼re Þ r00 r2e ðpi r00 Þr20
The radial displacement for a circular tunnel can be worked out based on the elasto-plastic theory. The strains in the plastic region are composed of elastic and plastic strain, and are expressed as Eqs. (27) and (28). The superscripts e and p represent the elastic and plastic parts, respectively. By considering compressive strains and radially inward displacements to be positive, the relationship between strain and displacement at any point in the soil-mass can be written as follows:
qffiffiffiffiffi kw ¼ 0
ð35Þ
In order to evaluate the integral in the above equation, expressions for eer and eeh can be obtained by following equation (Brady and Brown, 1993):
pi r0rðr¼r
eÞ r 2e r 20
r 20 r 2e
Here C ¼ , D¼ , and t is the r 2e r 20 Poisson’s ratio of the soil-mass. By substituting Eqs. (37) and (38) into Eq. (34), and then substituting the resulting expression for f(r) into Eq. (36), the integration of Eq. (36) can be calculated mathematically. The expression for the radial displacement in the plastic region can be obtained as follows (Sharan, 2003):
ur ¼
Q ¼ f ðrr ; rh Þ ¼ rh kw rr 2c
r
3.2. Theoretical solutions for displacement
dur er ¼ dr u eh ¼ r r
Z
re
er ¼ eer þ epr eh ¼ eeh þ eph
re ðrv o rrðr¼re Þ Þ 2G
where G is the shear modulus of the soil-mass. Eqs. (33)–(35) lead to the following expressions for the radial displacement:
R20
1 1 0 1k re ¼ r0 2r 0 þ a þ ar pi þ ar 1 þ k 1þk Z re 1 1 þ cw ir ðnÞdn A½logðr ec Þ B 1 þ k ðv 1Þ R0
Z r e k 11 Z r0 r cw 1kr 1kr þ 1k n i n i r ðnÞdn r ðnÞdn r re R0 R0
ð34Þ
Eq. (33) can be solved by using the following boundary condition for the radial displacement, urðr¼re Þ at the elasto-plastic interface (Brady and Brown, 1993).
ur ¼ r kw
Finally, at the interface between the plastic and elastic regions, the radial stress calculated in the plastic region must be identical to that in the elastic region. Consequently, Eq. (15) should be equal to Eq. (25) since the radial stress should be continuous over the boundary. The following Eq. (26) of the radius of the plastic zone, re can be derived as follows:
eer þ kw eeh ¼ f ðrÞ
urðr¼re Þ ¼
where
Z
ð33Þ
where
r 0 2
The following Eq. (25) is derived from Eqs. (23) and (24) and Mohr–Coulomb yield criterion at the stress state in the elastic region.
r0re ¼
dur ur þ kw ¼ f ðrÞ dr r
i 1 kw h k þ1 k 1 Cð1 2tÞ rew r kw þ1 D r ew r kw 1 r 2G r kw e þ urðr¼re Þ r
ð39Þ
The radial displacement urðr¼r0 Þ at the opening surface r = r0 is given by Eq. (40).
urðr¼r0 Þ ¼
i 1 kw h k þ1 k þ1 k 1 k 1 r 0 Cð1 2tÞ r ew r 0w D rew r0w 2G kw re ð40Þ þ urðr¼re Þ r0
The ground reaction curve is estimated through theoretical solution for the cases in which the cover depth of tunnel and the water height are 10 times diameter of tunnel in Fig. 4. In this paper, nondilating soil-mass is utilized (W = 0). As shown in Fig. 10, the ground reaction curve with consideration of seepage force is bigger than the ground reaction curve in dry condition. This is due to additional seepage forces to the initial effective overburden pressure. 3.3. Elastic stress induced by seepage forces In this study, the superposition concept is used to determine the effective stress considering seepage. In order words, the effective stress including seepage in elastic region is a combination of the effective stress in dry condition and the effective stress only considering seepage. The theoretical solution in elastic region only by seepage is compared with studies of other researchers and estimated. Muir Wood (1975) states that the solution of elastic stress by seepage is as follows:
Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324
321
where ho is the water depth from the groundwater table to tunnel
100
2
o . axis level, and T ¼ 1 þ 4h r2
On the other hand, Bobet (2003) proposes the stress solution by using a far field pore water pressure, uf for the elastic ground mass.
Dry condition
80
Seepage condition
0 r
r
60
pi(%) 40
20
r0h ¼
0
0.1
0.2
0.3
0.4
0.5
Fig. 10. The ground reaction curve (C/D = 10, H/D = 10).
qðr 2o ln r o r 2 ln rÞ 2kð1 mÞr2
q r 2 ln r o þ r 2 ln r þ r2 ¼ 1 o k 2ð1 mÞr 2
rr ¼
ð41Þ
rh
ð42Þ
In this case, ho is the water depth from the groundwater table to tunnel axis level, and q is flow rate into the tunnel per unit length of tunnel. Muir Wood (1975) derives theoretical solutions from the flux of flow into a tunnel. He insists that the stresses by water might add to ground load stresses to establish the overall condition of loading supported by the tunnel. Also, a lining with permeability relatively lower than that of the ground reduces the flow per unit length in Eqs. (41) and (42), thus leading to the reduction of ground loading, while the fraction of water load carried directly on the lining compensates for it. His solution contains a mistake in the boundary conditions. Curtis et al. (1976) modified that by changing boundary condition as follows:
rr ¼ ln R0 R20 r 20
r 20 R20
r 20
ln
! R0 q 2kð1 mÞ r0
r20 R20 q ln Rr00 2kð1 mÞ r2
q lnðrÞ 2kð1 mÞ
ð43Þ
! R0 q ¼ ln R0 ln 2kð1 mÞ r0 r20 R20 R20 r 20 q ln Rr00 2kð1 mÞ r 20 R20 q lnðrÞ q q þ 2kð1 mÞ 2kð1 mÞ k r2 r 20
ð44Þ
9 2 ln r þ ½ð1 2mÞT 2ð1 mÞ ln ðr2 r2o Þ> = 2 1 ro ro T o 1 þ 2 > 2ð1 mÞ > r ln T ; : 8 > <r 2
ð45Þ
r0h ¼ cw ho
8 <
1 r 2o þ1 2 2ð1 mÞ : r
h
i 9 2 ðr 2 r 2 Þ o þ 2m ln r2 T o = 2 ln rro ð1 2mÞ 1 þ 4h r2 o
ln T
ln Rro þ 1 2m
ð48Þ
As discussed above, Fig. 11 shows the elastic stress by the seepage forces due to excavation of tunnel below the groundwater table. The seepage stress, r0ei in the diagram is normalized with the hydrostatic pressure at the tunnel crown, Dpw and the radial distance is normalized with tunnel diameter, D. The theoretical solution by Muir Wood (1975) is illustrated in Fig. 11a with the correction by Curtis et al. (1976) shown in Fig. 11b. Fig. 11b shows the theoretical solution by Stern (1965), Fernandez and Alvarez (1994), and Curtis et al. (1976) together, and compares each to the others. Each of these solutions is very similar at the tunnel perimeter, but they have a tendency to increase in difference with distance from the tunnel due to boundary conditions. This is because Fernandez and Alvarez (1994) and Curtis et al. (1976) used the infinite boundary condition, r = 1, r0h ¼ 0 and Stern (1965) used the finite boundary, r = R0, r0h ¼ 0. However, these differences can be negligible in magnitude. Therefore, the usage of finite boundary is reasonable. Fig. 11c shows the theoretical solution by Bobet (2003), which proves to be different from others’ results, because the far field pore water pressure is considered as external load. According to the summary above, it is reasonable to apply the theoretical solution by Stern (1965) to estimate the seepage stress in elastic region. 4. Ground reaction curve considering seepage forces
Fernandez and Alvarez (1994) suggest the solution of elastic stress due to seepage under steady-state flow as follows:
r0r ¼ cw ho
ð47Þ
" # 2 uf Ro þ1 þ r 2ð1 mÞ
ln Rroo
r 2 uf o 1 ð1 2mÞ 2ð1 mÞ r
0.6
ur(m)
rh
1 2m r 2o uf 2 2ð1 mÞ Ro r 2o
0
þ
" # 2 ln Rro uf 1 2m r 2o Ro ¼ uf 1 þ 2 2ð1 mÞ Ro r 2o r 2ð1 mÞ ln Rr o o
r 2 uf o 1 þ ð1 2mÞ 2ð1 mÞ r
; ð46Þ
To validate the theoretical solution considering seepage forces, the ground reaction curves achieved by the numerical analysis and theoretical solutions are compared with each other and estimated. To verify the application of numerical analysis modeling used in this study, the ground reaction curves of the theoretical solution, which was suggested by Sharan (2003), and computed by numerical analysis in dry condition are compared. Based on this verification, the theoretical solutions of ground reaction curve considering seepage forces are evaluated and the applicability of theoretical solutions induced in this study is suggested. 4.1. General conditions for numerical analysis The program used in numerical analysis is PENTAGON 2D (Emerald Soft Consulting Co., 1998). It is a finite element analysis program for solving two-dimensional continuum problems and seepage problem. Both of the associated flow rule and non-associated flow rule are adopted when plastic strain is present after the material yields. The soil-mass is assumed to be governed by a
322
Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324 Table 4 Cases for analysis under dry condition (H/D = 0).
a
C/D
Dry condition
′ σ rei
5.0
10.0
15.0
Case 1
Case 2
Case 3
Δpw
σ θ′ei
σ ei′
Table 5 Cases for analysis when considering seepage forces.
Δpw
Δpw
H/D
C/D 5.0
2.5 5.0 7.5 10.0
Δpw
7.5 21 22 23 24
Case Case Case Case
10.0 31 32 33 34
Case Case Case Case
41 42 43 44
the tunnel depth, C and groundwater table, H as shown in Tables 4 and 5. The boundary conditions for seepage analysis are shown in Fig. 12. To carry out the seepage analysis, a total head is applied at the outer boundary before the tunnel excavation, and only the elevation head is applied with its pressure head being zero at the cross-section of the tunnel after excavation. To get the ground reaction curve by the numerical analysis, the internal pressure acting on the tunnel decreases with the ten steps by ten percent, and the radial displacements of the tunnel are found at each step.
b
σ ei′
Case Case Case Case
σ θ′ ei Δpw
4.2. Procedure for evaluating the ground reaction curve by theoretical solution
′ σ rei Δpw
The ground reaction curve by theoretical solution can be obtained from the equations derived before, and the general procedure is as follows: first, calculate hydraulic gradient distribution using numerical analysis in the case of seepage condition (seepage analysis is not needed under dry condition); assume the value of internal pressure, pi; calculate the outer boundary of plastic region, re and radial displacement, ur at the cross-section of tunnel by using Eqs. (26) and (40), respectively; and complete the ground reaction curve, which represents the relationship between internal pressure and radial displacement by repeating the above process with newly assumed values of internal pressure.
c ′ σ rei Δpw σθe ' ′ σ θ ei Δpw Δpw
σ re '
Δpw
σ ei′ Δpw
4.3. Comparison between analytical solution and numerical analysis
Fig. 11. The theoretical solution of seepage stress in elastic region. (a) The theoretical solution of seepage stress in elastic region by Muir Wood (1975). (b) The theoretical solutions of seepage stress in elastic region. (c) The theoretical solution of seepage stress in elastic region by Bobet (2003).
linear Mohr–Coulomb yield criterion. The properties of the ground materials used in the analysis are shown in Table 3. A circular tunnel with a diameter of 5.0 m, as shown in Fig. 4, is analyzed, and the ground reaction curve is calculated by changing
To verify the feasibility of the numerical analysis, the ground reaction curves from the theoretical solution and numerical method under dry conditions (H/D = 0) suggested by Sharan (2003) are obtained and compared with each other. Fig. 13 shows the ground reaction curves at tunnel crown under dry conditions by both theoretical solution and numerical analysis in the case that the cover depths of the tunnel, C were the 5, 10, and 15 times the diameter of tunnel, D respectively. The ground reaction curves obtained by both methods are in close agreement. Therefore, the numerical analysis modeling carried out in this study is appropriate. To verify the theoretical solution considering seepage forces induced in this study, the ground reaction curves by the theoretical solution are compared with those by numerical analysis. For these cases, the cover depths of tunnel, C are the 5 times, 7.5 times and
Table 3 Properties of ground for analysis. Soil type
Submerged unit weight, csub (kN/m3)
Cohesion, c (kN/m2)
Internal friction angle, / (deg)
Dilation angle, w (deg)
Elastic modulus, E (kN/m2)
Poisson’s ratio
Weathered soil
4.9
9.81
35.0
0.0
49,000
0.30
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Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324
100
pi (%)
Theoretical solution-H/D=2.5
90
Theoretical solution-H/D=5.0
80
Theoretical solution-H/D=7.5
70
Theoretical solution-H/D=10.0 Numerical solution-H/D=2.5
60 50
Numerical solution-H/D=5.0
40
Numerical solution-H/D=7.5 Numerical solution-H/D=10.0
30 20 10 0 0
0.2
0.4
0.6
0.8
1.0
Radial displacement (m) Fig. 14. The ground reaction curves with consideration of seepage force (C/D = 5.0).
100 Fig. 12. Boundary conditions for numerical analysis.
100
Theoretical solution-C/D=5
90 80 70 60
pi (%)
pi (%)
Theoretical solution-H/D=5.0
80
Theoretical solution-H/D=7.5
70
Theoretical solution-H/D=10.0 Numerical solution-H/D=2.5
60
Theoretical solution-C/D=10
50
Numerical solution-H/D=5.0
Theoretical solution-C/D=15
40
Numerical solution-C/D=5
30
Numerical solution-H/D=7.5 Numerical solution-H/D=10.0
20
Numerical solution-C/D=10
50
Theoretical solution-H/D=2.5
90
10
Numerical solution-C/D=15
0
40
0
0.2
0.4
0.6
0.8
1.0
Radial displacement (m)
30 20
Fig. 15. The ground reaction curves with consideration of seepage (C/D = 7.5).
10
100
0 0
0.03
0.06
0.09
0.12
Radial displacement (m) Fig. 13. The ground reaction curves under dry condition according to the C/D variation.
10 times diameter of tunnel, D with the variation of groundwater table, H from the 2.5 times to 10 times diameter of tunnel. The ground reaction curves are drawn in Figs. 14–16 respectively. It is found that the results of the numerical method are almost similar with those of the theoretical solution. The theoretical solutions of ground reaction curve with consideration of seepage forces are verified with numerical analysis. However, the differences between the results of theoretical solution and numerical analysis can be bigger than those in the case of dry conditions in the same cases. The reason for this difference is due to the method in which seepage is dealt with in the theoretical solution with the consideration of seepage. As mentioned before, the groundwater would flow non-radially in most cases. This means that seepage considered in theoretical solution can be different from the actual flow. For the analysis by theoretical solution, the only radial component of hydraulic gradient distribution is calculated by the numerical analysis, and the seepage forces are then added by integration of the radial component of hydraulic gradient to the theoretical solution. This means that the whole seepage may not be reflected in the solution in some conditions. In case of the numerical method, however, after the hydraulic gradient is calculated by the total head loss at all meshes from groundwater table to tunnel wall, all the seepage forces are accumulated from each mesh acting on the tunnel, and are applied to the analysis. Therefore, this discrepancy between the two cases may account for the differences in the results,
pi (%)
Theoretical solution-H/D=2.5
90
Theoretical solution-H/D=5.0
80
Theoretical solution-H/D=7.5
70
Theoretical solution-H/D=10.0 Numerical solution-H/D=2.5
60 50
Numerical solution-H/D=5.0
40
Numerical solution-H/D=7.5 Numerical solution-H/D=10.0
30 20 10 0 0
0.2
0.4
0.6
0.8
1.0
Radial displacement (m) Fig. 16. The ground reaction curves with consideration of seepage (C/D = 10.0).
as shown in Fig. 14 (H/D = 10.0, C/D = 5.0) and Fig. 16 (H/D = 2.5, C/ D = 10.0). These show the bigger discrepancies when compared with other cases. The first case is twice higher groundwater table than cover depth (H/C = 2) and the second one is the case of four times lower groundwater table than cover depth (H/C = 0.25). The fact that there are big differences in the case of too high or too low water height relative to cover depth shows the limit of usage of the theoretical solution proposed in this study. 4.4. The applicability of the theoretical solution of ground reaction curves As discussed in the previous chapter, according to the comparison of the theoretical solution and numerical analysis, the theoretical solution of this study can be applied in the cases where the ratio of
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Y.-J. Shin et al. / Tunnelling and Underground Space Technology 25 (2010) 315–324
water height and cover depth, H/C is ranged between 0.25 and 2 (0.25 6 H/C 6 2). Moreover, the theoretical solution in this study is valid for radial flow. The ground water flow is essentially radial when the ratio of water height and diameter of tunnel, H/D is over 10, in cases where the water height equals to cover depth (Fernandez and Alvarez, 1994). These are applicable limits of the theoretical solution derived when considering seepage in this study. 5. Conclusions In this study, the effect of seepage force on tunnels is investigated using analytical and numerical methods. Theoretical solutions including seepage forces are proposed, and validated using numerical methods. It is found that the influence of seepage force on the ground reaction curve is significant. The conclusions reached and the observations provided contribute to a better understanding of the behavior of tunneling below the groundwater table. The results obtained from this study can be summarized as follows: (1) The range for calculating seepage forces is defined as the distance where the radial hydraulic gradient becomes to zero. This is because the distinct failure plane does not occur in tunnels different from earth retaining walls and tunnel faces. When the groundwater table is below ground surface, the seepage pressure ratios show the range of 85–90% of the hydrostatic pressure at a tunnel crown. When the groundwater table was higher than the ground surface, the seepage pressure ratios is over 95% of the hydrostatic pressure at tunnel crown. (2) It is shown that the flow of groundwater into tunnels results in significant increase in the radial displacements of the tunnel wall due to seepage force. (3) In comparison with dry conditions, the existence of flow into a tunnel causes seepage force and consequently has reduced the arching effect during excavation and large displacements. (4) The theoretical solution of ground reaction curve with consideration of seepage forces is verified with numerical analyses. Through the comparison of theoretical solution and numerical analysis, it is found that the theoretical solution in this study is applicable when the ratio of water height and cover depth, H/C ranges from 0.25 to 2, and when the ratio of water height and tunnel diameter, H/D is over 10 for an equal ratio of water height and cover depth (H/C = 1). Acknowledgement This paper was supported by the Underground Space Construction Technology Center under the Ministry of Construction and Transportation in Korea (Grant C04-01).
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