the ground reaction curve due to tunnelling under drainage condition

7/27/2019 The Ground Reaction Curve due to Tunnelling under Drainage Condition

http://slidepdf.com/reader/full/the-ground-reaction-curve-due-to-tunnelling-under-drainage-condition 1/8

The Ground Reaction Curve due to Tunnelling under Drainage Condition

Young-jin Shin1, Byoung-min Kim

2, Shin-in Han

3, In-mo Lee

4, and Daehyeon Kim

5

1School of Civil Engineering, Purdue University, West Lafayette: [email protected]

2Underground Space Construction Technology Centre, Seoul, Korea

3Department of Civil Engineering, Korea University, Seoul, Korea

4Department of Civil Engineering, Korea University, Seoul, Korea: [email protected]

5Indiana Department of Transportation, West Lafayette

ABSTRACT: When a tunnel is excavated below the groundwater table, water flows

into the excavated wall of tunnel and seepage forces are acting on the tunnel wall.

Such seepage forces significantly affect the ground behavior. The ground response totunnelling is understood theoretically by the convergence-confinement method, which

consists of three elements: longitudinal deformation profile, ground reaction curve,

and support characteristic curve. The seepage forces are likely to have a strong

influence on the ground reaction curve which is defined as the relationship between

internal pressure and radial displacement of the tunnel wall. In this paper, seepage

forces arising from the ground water flow into a tunnel were estimated quantitatively.

Magnitude of seepage forces was determined based on hydraulic gradient distribution

around tunnel. To estimate seepage forces, different cover depths and groundwater

table levels were considered. Using these results, the theoretical solutions for theground reaction curve (GRC) with consideration of seepage forces under steady-state

flow were derived.

INTRODUCTION

When a tunnel is excavated below the groundwater table, groundwater may flow

into the tunnel and, consequently, seepage forces may develop in the ground seriously

affecting the behavior of the tunnel. Ground response to tunnelling can be understood

theoretically by the convergence-confinement method. This method is based on the

principle for which a tunnel is stabilized by controlling its displacements afterinstallation of a support near the tunnel face. The convergence-confinement method is

based on three elements: the longitudinal deformation profile, the ground reaction

curve, and the support characteristic curve. The longitudinal deformation profile

assuming no support shows the radial displacement of the tunnel cross-section in the

longitudinal direction from the tunnel face. The support characteristic curve describes

394

Copyright ASCE 2008 GeoCongress 2008GeoCongress 2008 o

m a

s c e l i b r a r y . o r g b y K a i s t K o r e a A d

v a n c e d I n s t . O f o n 1 0 / 0 8 / 1 2 . F o r p e r s o n a l u s e o n l y .

N o o t h e

r u s e s w i t h o u t p e r m i s s i o n .

C o p y r i g h t ( c ) 2 0 1 2 .

A m e r i c a n S o

c i e t y o f C i v i l E n g i n e e r s .

A l l



the increasing pressure that acts on the supports as the radial displacement of the

tunnel increases. Lastly, the ground reaction curve shows the increasing trends of

radial displacement as the internal pressure of the tunnel decreases. Tunnelling belowthe ground water table induces additional seepage stresses (Shin et al., 2007), and the

seepage forces are likely to have a strong influence on the ground reaction curve.

Previous studies on the ground reaction curve by Stille (1989), Wang (1994),

Carranza-Torres( 2002), Sharan (2003), and Oreste (2003) did not consider seepage

forces. The effects of seepage forces on the tunnel face or the support system were

studied by Muir Wood (1975), Curtis (1976), Atkinson (1983), Schweiger (1991),

Fernandez and Alveradez (1994), Fernandez (1994), Lee and Nam (2001), Bobet

(2003), Shin et al. (2005). 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 been suggested.In this study, based on these previous studies, the theoretical solutions of the

ground reaction curve considering seepage forces due to groundwater flow under

steady-state flow were derived.

THEORETICAL SOLUTION OF GROUND REACTION CURVE WITH

CONSIDERATION OF SEEPAGE FORCES

Theoretical solution 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, as indicated in Figure 1.

1 3 ( 1)k k a = + (1)

Here1

indicates the major principal stress,3 is the minor principal stress,

2tan (45 )2

k

= + ,tan

ca

= , where k and a are the Mohr-Coulomb constants, c is

the cohesion, and is the friction angle.

c

c

a

a

1

' 0

i p

or er

'

0

'

0 '

0

FIG. 1. Elasto-plastic model based on

Mohr-Coulomb yield criterion

FIG. 2. Circular opening in an infinite

medium

Figure 2 shows a circular opening of radius 0r with 0 1k = in an infinite soil-mass

GEOCONGRESS 2008: GEOSUSTAINABILITY AND GEOHAZARD MITIGATION 395


m a



N o o t h e


C o p y r i g h t ( c ) 2 0 1 2 .

A m e r i c a n S o


A l l



For the plastic region, (1) can be modified as follows:

( 1)r r r r k k a = +

(9)

where 2tan (45 )

2

r r k

= + ,

tan

r r

r

ca

= , r

k and r a are the Mohr-Coulomb constants,

r c is the cohesion, and r is the friction angle in the plastic region.

Substituting (9) into (8) and solving it with the boundary conditions r i p = at

0r r = . Then, the radial and circumferential effective stresses in the plastic region are

as follows (Shin et al., 2007):

( ) ( ) ( )0

0 0

1

1 10

1[ ]

r

r r

r

k r r

k k w

rp i r r r r k R R

r p a a i d i d

r r

= +

(10)

( ) ( ) ( )0

0 0

1

1 10

1[ ]

r

r r

r

k r r

k k w

p r i r r r r r k R R

r k p a a k i d i d

r r

= +

(11)

where, 0 R is the distance from ground to the center of tunnel.

In this equation, i p is all the support pressure developed by in situ stress and

seepage. Subscripts rp and p are the radial and tangential effective stresses in the

plastic region, respectively.

In order to estimate the effective stress in the elastic region, the superposition

concept is used. As shown in Figure 4, the effective stress considering the seepage

force can be assumed as a combination of the solution of the equilibrium equation for

the dry condition and the effective stress only considering seepage.

FIG. 4. Concept of superposition in elastic region.

The Kirsch solutions are applied to solve the effective stresses in the elastic region

under dry condition (Timoshenko and Goodier, 1969).

For the seepage condition, Stern (1969) suggests effective stresses in the elastic

region with consideration of the seepage force as follows:

( ) ( ) ( ) ( ) ( )2

1 2 1

2log 14 2 2 1 2 1rei

C A B v

r I r J r r v v

= (12)

( )( )

( )( )

( )2

1 2 12log 1

4 2 2 1 2 1ei

C A B vr I r J r

r v v

= + +

(13)



m a



N o o t h e


C o p y r i g h t ( c ) 2 0 1 2 .

A m e r i c a n S o


A l l



where ( ) ( )0

r

w r R

I r i d = , ( ) ( )0

2

2

r w

r R

J r i d r

= , C, A and B are constants

defined by the boundary conditions.(12) and (13) can be solved by using the boundary conditions 0

rei = at 0r r = ,

0rei = at

0r R= , and 0ei = at

0r R= .

Here, subscript i represents the term related to seepage.

Consequently, the radial and tangential effective stresses with consideration of the

seepage forces in the elastic region can be obtained by the superposition of both of

solutions as follows:

( )

( )( )

( )( ) ( )

2 20 0 0 0 0

2 2 2 20 0 0 0

2

0 0

0 020 0 0

2 2

0 0

1 12log 1

1 2log 2log 1 1 2log 2log 1

2log 1 2 1

1 2log 2log 1 2 1 2 1

re

i

Z Z r

R r R r r R

r R r R

Z R v r I r J r p

R r R v v r

r R

= + +

+ +

+ + + +

(14)

( )

( ) ( ) ( ) ( ) ( )

2 20 0 0 0 0

2 2 2 2

0 0 0 0

2

0 0

0 020 0 0

2 2

0 0

1 12log 1

1 2log 2log 1 1 2log 2log 1

2log 1 2 1

1 2log 2log 1 2 1 2 1

e

i

Z Z r

R r R r r R

r R r R

Z R v r

I r J r p R r R v v r

r R

= + + ++ +

+ + + + + +

(15)

Here,( )

( )( )

( )0 0

0 0

2

2

0

1 2 1

2 1 2 1

r r w

w r r R R

v Z i d i d

v v r

= +

(16) is derived from (14) and (15) and the Mohr-Coulomb yield criterion at the

stress state in the elastic region.

( ) ( )( )

( )0

1 1 1 12 ( log )

1 1 1 1er e e

k a A r B I r

k k k v

= + +

+ + + (16)

Where,2

0 0 0

2 2

0 0

4

2log 2log 11

Z A

R r R

r R

=

++

0

2

0 0 0

2 2

0 0

4log

2log 2log 11

R Z B

R r R

r R

=

++

Finally, at the interface between the plastic and elastic regions,e

r r = , the radial

stress calculated in the plastic region must be identical to that in the elastic region.

Consequently, (10) should be equal to (16) since the radial stress should be continuous

over the boundary. The radius of the plastic zone, er , can be derived as follows:

( )( )

( )

( )

( ) ( )

0

0

0 0

1

1

0

0

1 1

1

1 1 1 12 (

1 1 1 11log )

[ ]

e r

er r

r

r k

r w r R

e ec

i r r r

k k wr r k R R

e

k a a i d

k k k vr r A r B

p a

i d i d r

+ + +

+ + +

= + + ! "

(17)

GEOCONGRESS 2008: GEOSUSTAINABILITY AND GEOHAZARD MITIGATION398


m a



N o o t h e


C o p y r i g h t ( c ) 2 0 1 2 .

A m e r i c a n S o


A l l



Theoretical solution for displacement

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

strains, and are expressed as Eqn. (18) and (19), respectively. 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 a soil-mass can be written as follows:e p

r r r = +(18)

e p

= +(19)

r r

du

dr

=

(20)

r u

r

=

(21)

The plastic strain can be represented by using the plastic flow rule. When the

volume expansion effect is important in plastic strain, generally the non-associated

flow rule is valid; otherwise, the associated flow rule is valid. The plastic potential

function, Q , when using non-associated flow rule, is as follows:

( , ) 2 0r r Q f k c k

# # = = =

(22)

where 1 sin1 sin

k # # #

+=

, the parameter # is the dilation angle.

The plastic parts of radial and circumferential strains can be related as follows:

r

p pk

# = (23)

Eqn. (20) ~ (23) lead to the following differential equation.

( )r r du uk f r

dr r # + =

(24)

where ( )e

r k f r # + =e (25)

Eqn. (24) can be solved by using the following boundary condition for the radial

displacement, ( )er r r u=

, at the elasto-plastic interface (Brady and Brown, 1993).

( ) ( )( )2e er r r vo r r r

bu

G

= =

=

(26)

where G is the shear modulus of the soil-mass.

Eqn. (24) - (26) lead to the following expressions for the radial displacement:

( )( )e

e

k r k k e

r r r r r

r u r r f r dr u

r

#

# #

=

= +

(27)

In order to evaluate the integral in the above equation, expressions for er and e

can be obtained by the following equation (Brady and Brown, 1993):

2

1[(1 2 ) ]

2

e

r

DC

G r % = +

(28)



m a



N o o t h e


C o p y r i g h t ( c ) 2 0 1 2 .

A m e r i c a n S o


A l l



2

1[(1 2 ) ]

2

e DC

G r % =

(29)

Here2 2

( ) 0 0 0

2 2

0

( ) ( )er r r e i

e

r p r C

r r

= =

,

2 2( ) 0

2 2

0

( )ei r r r e

e

p r r D

r r

==

, and % is the

Poisson`s ratio of the soil-mass.

Eqn. (26) can be solved by using Eqn. (28) and (29). The expression for the radial

displacement in the plastic region at the opening surface 0r r = is given by Eqn. (30).

0

1 1 1 1

( ) 0 0 0 ( )

0

1[ (1 2 )( ) ( )] ( )

2 e

k k k k k k er r r e e r r r

r u r C r r D r r u

G r

# # # # # # % + +

= == +

(30)

The ground reaction curve is estimated by using the theoretical solutions for the

cases in which the cover depth of the tunnel, C , and water height, H , are 10 timesthe diameter of tunnel, D . As shown in Figure 5, the ground reaction curve with

consideration of seepage force shows larger radial displacement than the ground

reaction curve for the dry condition; this result means that there is no ground water

when the cover depth of the tunnel, C , is 10 times the diameter of the tunnel, D .

This is due to the fact that even if the effective overburden pressure can be decreased

by the arching effect during tunnel excavation, seepage forces still remain.

FIG. 5. The ground reaction curve ( / 10C D = , / 10 H D = )

CONCLUSIONS

The flow of groundwater has a significant effect on the radial displacement of a tunnel

wall. While the effective overburden pressure is reduced slightly by the arching effect

during tunnel excavation, seepage forces still remain. Therefore, the presence of

groundwater induces larger radial displacements of the tunnel wall than those in the

case of dry condition.

GEOCONGRESS 2008: GEOSUSTAINABILITY AND GEOHAZARD MITIGATION400


m a



N o o t h e


C o p y r i g h t ( c ) 2 0 1 2 .

A m e r i c a n S o


A l l



ACKNOWLEDGMENTS

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. and Mair, R.J. (1983). "Loads on leaking and watertight tunnel lining,

sewers and buried pipes due to groundwater." Geotechnique, Vol. 33(3): 341-344.

Brady, B.H.G. and Brown, E.T. (1993). “Rock mechanics for underground mining.”

London; Chapman and Hall.

Carranza-Torres, C. (2002). "Dimensionless Graphical Representation of the ExactElasto-plastic Solution of a Circular Tunnel in a Mohr-Coulomb Material Subject

to Uniform Far-field Stresses." Rock Mechanics and Rock Engineering, Vol. 36(3):

237-253.

Fernandez, G. and Alvarez, T.A. (1994). "Seepage-induced effective stresses and

water pressures around pressure tunnels." J. Geotechnical Engineering, Vol.

120( 1): 108-128.

Lee, S.W., Jung, J.W., Nam, S.W. and Lee, I.M. (2007). "The influence of seepage

forces on ground reaction curve of circular opening." Tunnelling and Underground

Space Technology, Vol. 21: 28-38.

Oreste, P.P. (2003). "Analysis of structural interaction in tunnels using theconvergence-confinement approach.", Tunnelling and Underground Space

Technology, Vol. 18: 347-363.

Schweiger, H.F., Pottler, R.K. and Steiner H. (1991). "Effect of seepage forces on the

shotcrete lining of a large undersea cavern." Computer Method and Advances in

Geomechanics, Beer, Booker & Carter (eds), pp. 1503-1508.

Sharan, S.K. (2003). "Elastic-brittle-plastic analysis of circular openings in Hoek-

Brown media." International J. of Rock Mechanics and Mining Sciences, Vol. 40:

817-824.

Shin, J.H., Potts, D.M. and Zdravkovic, L. (2005). "The effect of pore-water pressureon NATM tunnel lingings in decomposed granite soil." Canadian Geotechnical J.,

Vol. 42: 1585-1599.

Shin, J.H., Lee, I.M. and Shin, Y.J. (2007). "Seepae-induced Stress due to Tunnelling

under Drainage Condition.", Canadian Geotechnical J., (Submitted).

Stille, H., Holmberg, M. and Nord, G. (1989), "Support of Weak Rock with Grouted

Bolts and Shotcrete", International J. of Rock Mechanics and Mining Sciences &

Geomechanics Abstracts, Vol. 26, No. 1, pp. 99-113.


a s c e l i b r a r y . o r g b y K a i s t K o r e a A d


N o o t h e


C o p y r i g h t ( c ) 2 0 1 2 .

A m e r i c a n S o


A l l

the ground reaction curve due to tunnelling under drainage condition

Documents