Tunnels in Cohesion Less Soil

International Symposium on Underground Excavation and Tunnelling 2-4 February 2006, Bangkok, Thailand 

Effect of Groundwater on Tunnel Stability Chatawut Chanvanichskul1, Takeshi Tamura1 1

Department of Civil and Earth Resources Engineering, Kyoto University, Japan

ABSTRACT

The stability of tunnel is one of the most important subjects in the tunnel constructions, especially when the groundwater table locates above the tunnel. However, there have been few researches with respect to the effect of the the groundwater on the the stability of the tunnel. In this paper, the fundamental studies on the effect of the groundwater on the stability of tunnel are numerically carried out using the rigid-plastic finite element method (RPFEM) based on the upper bound theorem. The effect of the groundwater is represented by applying the water pressure as a force on the ground. The twodimensional cross-section tunnel embedded in the cohesive-frictional ground is mainly considered in this paper. The unlined tunnel is considered first by varying the groundwater table. When groundwater table is higher than the tunnel springline, the stability of tunnel driven in the sandy ground is abruptly decreased from the tunnel stability in the lower groundwater groundwater table than the tunnel springline. Second, the lined tunnel is taken into account by varying the patterns of lining and the groundwater table. According to the results, when the groundwater table is high the reinforcement at the invert of tunnel plays an important role to stabilize the overall stability of tunnel.

1. INTRODUCTION

The studies on the evaluation of the stability of tunnel face were done by several authors in the last thirty years. For instance, in the case of tunneling in cohesive soil, Broms and Bennermark(1967), Mair(1979), Davis et al.(1980), al.(1980), Takemura (1990), Sloan et al.(1994) studied this issue by means means of  theoretical approach and experimental experimental approach. approach. In the case of tunneling in frictional soil, numerous theoretical studies were done by Atkinson and Potts(1977), Leca and Dormieux(1990), Tamura et al.(1999). Chambon and Corte(1994), Mashimo(1998) Mashimo(1998) have done the experimental experimental studies to evaluate the tunnel stability in frictional soil. However, these works are only valid for tunnels driven driven in either dry soil or fully saturated soil. No effects of the groundwater were taken into account. So far, there are very few studies considering the effect of groundwater on the stability of the tunnel face such as Buhan et al.(1999), Lee and Nam(2001), Droniuc et al.(2004). They presented the numerically method to determine the tunnel pressure required to maintain the tunnel stability when the tunnel is excavated under groundwater groundwater table by combining the effects of gravity and seepage forces. Konishi and Tamura et al.(2003) proposed the method to include the groundwater into the rigid-plastic finite element method and studied fundamentally the effect of groundwater on the simple geotechnical problems and simple tunnel problems. But, this method can analyze only the problem with nondilatant condition in soil, i.e. non-associated flow rule. In this paper, the new method which can analyze both associated and non-associated flow rules, to include the groundwater into the rigid-plastic finite element method is briefly described. Some fundamental studies of the effect of groundwater on the tunnel stability are carried out. The two dimensional cross-section cross-section tunnel embedded embedded in cohesive-frictional soil is mainly considered. considered. Not only the effect of groundwater on the unlined tunnel stability but also the lined tunnel stability is evaluated.

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2. RPFEM WITH CONSIDERATION OF GROUNDWATER

The rigid-plastic finite element method (RPFEM) based on upper bound theorem on plasticity is obtained through minimizing the rate of internal plastic energy dissipation  D (ε &ij ) with respect to the kinematically admissible velocity field under several linear constraint conditions (Tamura et al.(1984)) as:

∫

Minimize  D(ε &ij ) = σ ij ε &ij dV  subject to constraint conditions.

(1)

V 

where ε &ij is the plastic strain rate and σ ij is the stress tensor. V  denotes the volume of the considered region. RPFEM solves only the critical state or the moment when the whole region of the body begins to flow with a constant rate of deformation. In this paper, the stability of the tunnel is evaluated by determining the load factor which magnifies the acceleration of gravity g . Therefore, the actual collapse occurs under the 1g condition, if the obtained load factor

is smaller than 1. This evaluation method has a similar concept to that of 

the centrifugal test. At the critical state, not only the unit weight of soil γ  but also the pore water pressure  p is multiplied by the load factor

. To include the groundwater in the analysis, the

following three factors are taken into account: • Change in the total self-weight due to the change in the groundwater table; • Change in the effective stress due to the change in the total self-weight and pore water pressure; • Strength of the ground due to the change in the effective stress.

Figure 1. Calculations of unit weight and pore water pressure

2.1 Self-weight of soil

According to Figure 1(a), the unit weight of soil above the groundwater table is denoted by γ d  and the unit weight of soil under the groundwater table is denoted by γ s . The unit weight of the soil element

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through which the groundwater table passes as shown in Figure 1(b), is estimated as

γ  = (1 − α γ  )γ d  + α γ γ s

(2)

where the coefficient α γ  is defined by

α γ  =

 H c − y min

(3)

 y max − y min

in which  y max and  y min denote the maximum  y coordinate and the minimum  y coordinate of the element, respectively.  H c is the water head at the center of the element as shown in Figure 1(b). 2.2 Pore water pressure

The distribution of the total water head  H in the steady seepage state is determined in advance. The pore water pressure  p for each element can be calculated by averaging the pore water pressures of all 4 nodes as shown in the following equation.

⎧γ w ( H i − yi )  pi = ⎨ 0 ⎩

if   H i ≥  yi if   H i <  yi

(4)

where γ w is the unit weight of water.  yi and  H i denote, respectively, the y coordinate and the water head of the nodal point i . The pore water pressure  p of each element is calculated by 4

 p = ( ∑ pi ) / 4.

(5)

i =1

2.3 Effective stress

The effective stress σ ij′ is equal to the total stress σ ij subtracted by the pore water pressure  p , as follows:

σ ij′ = σ ij − µδ ij p

(6)

where δ ij is the Kronecker delta. The rigid plastic finite element method with the consideration of groundwater is carried out by considering the soil as non-homogenous soil i.e., the unit weight is varied by the groundwater table. The effective stress σ ij′ is considered in the stress-strain rate relation based on the Drucker-Prager yield criterion as follows:

σ ij′ = (G1 + G2 λ )

ε &ij e&

+ λ δ ij

(7)

in which λ  denotes the magnitude of the indeterminate stress component which is unknown variable.

e& is the magnitude of the strain rate which can be written as

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e& = ε &ij ε &ij

(8)

Furthermore, G1 and G2 are calculated by

G1 =

2 2{3(α  − α )λ  + (6α  + 1) k }

(1 + 6α α ) 6α  + 1 2

, G2

3 2α 

=

6α  + 1 2

.

(9)

α  and k  are non-negative material parameters for the Drucker-Prager material which associated with the Mohr-coulomb material parameter i.e., c and φ  . α  is the non-negative material parameter for non-associated flow rule and has a relationship with α  as: α  = r α 

(0 ≤ r ≤ 1)

(10)

where r is a non-associated flow rule coefficient indicating the dilatancy angle. By modifying Problem (1), the following basic equations for the rigid-plastic finite element method can be obtained:

•

 Equation of equilibrium:

σ ij , j +  f i = 0

(11)

where  f i is the body force vector. σ ij is the total stress tensor which can be obtained by combining Eqs. (6) and (7).

•

Kinematical constraint:

ε &kk  + G2 e& = 0

(12)

where ε &kk  denotes the volume change rate.

•

Constraint on the velocity on the basis of the upper bound theorem:

∫  f u& dV  + ∫ T u& dS  = 1 i

V

i

i

i

(13)

S 

where T i is surface traction and S is surface of the considered region.

u&i denotes the velocity

vector. The velocity u& of each node, the magnitude of indeterminate stress component λ  of each element and the load factor µ  are the unknown variables. The overall problem can be achieved by solving the system of equations composed of the above equations.

3. NUMERICAL ANALYSIS OF UNLINED TUNNEL 3.1 Description of the problem

The fundamental studies on the effect of the groundwater table on the unlined tunnel stability are carried out in this section. A 2D finite-element model for unlined cross-section tunnel and its region size are shown in Figure 2. The tunnel of diameter D is assumed to be excavated at the cover depth C  equal to D i.e., the cover-to-diameter ratio C/D is equal to 1. Only one half of the domain is considered in the finite-element model by making use of the symmetry about the tunnel centerline. The vertical velocities are only allowed at the left vertical boundaries i.e, centreline of tunnel. On the other hand, at

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the bottom boundary and right vertical boundary, both horizontal and vertical velocities are not allowed. The groundwater table is assumed to be horizontal and locates at the vertical distance  H measured from the tunnel springline Note that a notation + means the groundwater table locates above tunnel springline and the notation – denotes the case of which the groundwater table is below the tunnel springline. The soil cohesion is constantly equal to 30 kN/m 2. The angles of internal friction used in this study are 0o and 30o for purely cohesive soil and frictional soil, respectively. The unit weights of soil above the groundwater table and below the groundwater table are 15 kN/m 3 and 20 kN/m3, respectively.

Figure 2. Finite-element mesh for unlined tunnel

3.2 Results

Figure 3 shows the influence of the groundwater table on the stability of the unlined tunnel. The load factor µ  or the stability of the unlined tunnel in cohesive soil (i.e., φ  = 0 ) is not affected by the o

groundwater table

significantly. On the other hand, when φ  = 30 is considered, the groundwater o

table has the great effect on the stability of the tunnel. When the groundwater table is lower than the invert of tunnel, the groundwater table does not affect the tunnel stability but when the groundwater table is higher than the invert of  the tunnel the stability of tunnel starts to decrease and, moreover, when the groundwater table is higher than the springline of the tunnel the stability of  tunnel abruptly decreases. In addition, according to the same figure, when φ  = 30 is considered, the load factors obtained from the non-associated(r=0.0 and 0.5) and associated flow rules(r=1) are not significantly different each other. o

4. NUMERICAL ANALYSIS OF LINED TUNNEL 4.1 Description of the problem

The lining will be included in the analysis in order to investigate how the lining affects the stability of tunnel when groundwater table varies. Note that the word called 'lining'  does not mean only that is called the lining but also other reinforcement methods. Four patterns of lining are considered i.e., upper lining, lower lining, incomplete lining and complete lining. The Figure 3. Influence of groundwater table on the comparison between the upper and lower linings unlined tunnel stability unlined tunnel is considered first and the comparison between the incomplete and complete linings is considered later. The finite element models for all cases of the linings are illustrated in Figure 4. The darkly black elements are the lining elements. The boundary condition are the same as that of the previous section. The soil parameters are the same as those used in the unlined tunnel case. The lining is assumed to be purely cohesive material whose cohesion equals 3000 kN/m 2. The unit weight of lining is

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supposed to be independent of the groundwater table and equal to 24 kN/m 3. Also, in this case the cover-to-diameter ratio C/D is fixed to be unity.

(a) U

(b) Lower lining

er linin

(d) Complete lining

(c) Incomplete lining

Figure 4. Finite-element meshes for all cases of linings

4.2 Comparison between upper and lower linings

Figure 5 shows the influence of the groundwater on the stability of the tunnel when the upper and lower linings are considered. For the tunnel in cohesive soil (φ  = 0 ), both of the upper and lower o

linings do not increase the stability of the tunnel much. On the other hand, for the tunnel embedded in the cohesive-frictional soil (φ  = 30 ), both of these two lining patterns give some interesting effects o

on the stability of the tunnel. The upper lining strengthens the tunnel much when the groundwater table is lower than the springline. But if the groundwater table is higher than the springline, the upper lining does not improve the stability of the tunnel. For the lower lining, the lining improve the stability of the tunnel when the groundwater table is higher than the springline and does not improve the stability of the tunnel much when the groundwater table is lower than the springline. According to these results, we can conclude that the reinforcement at the invert of the tunnel plays the important role to improve the stability of the tunnel when the groundwater table is high. 4.3 Comparison between incomplete and complete linings

The influence of the groundwater on the stability of the tunnel for the incomplete and complete linings is shown in Figure 6. Apparently, the complete lining can strengthen the tunnel much both in cohesive

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and cohesive-frictional soils at any groundwater table. The incomplete lining does not affect the stability of the tunnel embedded in cohesive soil much. But, for the tunnel in the cohesivefrictional soil, the stability of the tunnel is improved much by the incomplete lining when the groundwater table is lower than the springline but if the groundwater table is higher than the springline the incomplete lining cannot improve the stability of the tunnel. These results are similar to the results obtained from the case of the upper lining. It can be again concluded that the reinforcement at the invert of the tunnel is vital for the stability of the tunnel. 5. CONCLUSIONS

The new method to include the effect of  groundwater in the rigid-plastic finite element method was described. The effect of  Figure 5. Influence of the groundwater for upper groundwater is represented by applying the and lower linings water pressure as a force to the ground. Only the two dimensional cross-section tunnel embedded in cohesive frictional soil was considered. For the unlined tunnel, the groundwater table has no much effect on the stability of the tunnel embedded in the cohesive soil but gives great effect on the stability of the tunnel embedded in the sandy soil. Especially, if the groundwater table is higher than springline of the tunnel, the stability of the tunnel decreases abruptly. This indicates that it is necessary to consider the existence of the groundwater during the excavation of the tunnel in sandy soil. For the lined tunnel driven in sandy soil, when the groundwater table is higher than the springline, the upper lining cannot improve the stability of the tunnel but the lower lining can improve the stability of tunnel. On the other hand, if the groundwater table is lower than the tunnel springline, the stability of the tunnel is Figure 6. Influence of the groundwater for improved by the upper lining but not by the incomplete and complete linings lower lining. Also, in the case of the incomplete lining, the stability of the tunnel does not increase when the groundwater table is higher than the springline, as the same as the result of the case of the upper lining. Conclusively, the above results suggest that the invert of the tunnel is necessary to be reinforced if the groundwater table is high.

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REFERENCES

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