convergence - confinement method

7/27/2019 Convergence - Confinement method

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International Symposium

on :Utilization of underground space in urban areas

6-7 November 2006, Sharm El-Sheikh, Egypt

ANALYSIS OF TBM TUNNELLING USING THE

CONVERGENCE-CONFINEMENT METHOD

Ashraf ABU-KRISHA

National Authority for Tunnels (NAT), Cairo, Egypt

Abstract:

The soft ground interaction in TBM (Tunnel Boring Machine) tunnelling is studied through the use of

the C-C (Convergence-Confinement) approach. The equations that characterize the behavior of thesupport (slurry pressures) are given together with a set of conceptual interaction schemes. As far as the

behavior of the support is concerned, reference is made to the ultimate limit state concept, which is

widely used in civil engineering. This approach is linked to the classical C-C method.

The static solution accounts for drainage and no-drainage conditions at the ground-liner interface.

Linear elasticity of the liner, non-linear elasticity of ground and plane strain conditions at any cross-

section of the tunnel are assumed. The analyses show that the stresses in the liner are slightly affected

whether there is drainage or not at the ground-liner interface. Hence, if the drainage conditions in the

tunnel are changed from full drainage to no-drainage or vice versa the stresses in the liners are

reasonably affected. Also, the displacements in the ground mass are changed significantly from

drainage to no-drainage conditions. Finally the ground reaction curve of the tunnel, which allows one

to analyze the structural interaction in TBM tunnel, is introduced.

Keywords: Convergence-Confinement, TBM Tunnelling, Soil interaction, drainage.

1. Introduction

During the excavation of tunnels in soft ground masses using conventional methods, several types of

temporary supports interact with the purpose of stabilizing the opening before the final lining is

completed. The study of the interaction between support structures can easily be carried out using the

convergence-confinement method. This method allows one to have a qualitative understanding of the

interaction phenomenon and helps operative choices. In more complex cases, it is necessary to use

numerical schemes; however, the computational effort required to analyze the results would need to be

intensive to produce a parametric analysis that is able to improve the final design.

The TBM tunnels are often found below the ground water table and thus the surrounding ground isfully saturated. The static loads on the liner depend on the drainage conditions at the contact between

the support and the ground. If there is no-drainage at the contact, the liner must support the pressures

generated by the water as well as the pressures generated by the ground. With full drainage the water

pressures become zero at the contact and the liner only needs to support the pressures from the ground.

Although the concept is clear, quantification of the loads on the support under any of the two drainage

conditions described is not trivial.

2. Convergence-confinement method

The convergence-confinement method for a circular tunnel in a hydrostatic stress field is given by

A.F.T.E.S., 1993 and Peila and Oreste, 1995, that shows the displacement and the load acting on the

support through the intersection of the ground reaction curve of the tunnel and reaction line of support,

Fig. 1. The reaction line can be defined on the basis of the following parameters:


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Stiffness k: ratio, in elastic conditions, between the pressure applied by the ground anddisplacement of the support;

Displacement of the excavation support uin, developed before the installation of the supports;

Pressure pmax, which induces the yielding of the support; Displacement of the tunnel liner umax, which causes the support to collapse;

Figure 1: The convergence-confinement method (p: internal tunnel pressure; u: radial displacement)

A support defined with the reaction line of Fig. 1 has an ideal elastic-perfectly plastic behavior. There

are often some uncertainties in the evaluation ofuin. As a first approximation, uin is evaluated using the

distance from the excavation face where the support is installed, the diameter of the tunnel and the

mechanical characteristics of the ground, Panet and Guenot, 1982; Panet 1995. The stiffness kof the

structure and the equilibrium loadpeq also influence the value ofuin. These uncertainties influence the

evaluation of the equilibrium point on the convergence-confinement curve. It is advisable to use a

parametric analysis for this calculation, varying uin in the interval of values that is considered to be

correct.

As the TBM tunnel is a three-dimensional problem, in order to take into account this effect in two-

dimensional analysis; the deconfinement ratio should be introduced. The

deconfinement ratio is a function of the distance x from the front of TBM,

the degree of soil plastification and the confinement slurry pressurepeq = i,as shown in Fig. 2. The effect of confinement for the soil excavation is

modeled via the relation:

where:

s is the soil deconfinement ratio and o = 0.9

o is the initial effective vertical stress at each point of the excavated surface.

3. Slurry pressure support

The support can be considered suitable when its safety factor concerning the collapse or yielding isgreater than an acceptable minimum and the displacements are lower than a given limit in relation to

)1(1

=

o

i

os


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the design criteria of the tunnel. The long-term performance of supports shall not be taken into

consideration when its strength exceeds the elastic limits. It is therefore, justifiable to impose the

elastic behavior of the lining through the definition of the safety factor that refers to the applied loads

p and not to the displacements u.

(2)

Figure 2: TBM tunnel excavation

4. Numerical analysis

There is a significant number of empirical, analytical and numerical procedures to obtain stresses in

the ground and in the liner for a tunnel, given the liner and the ground properties, ground water

pressure and drainage conditions at the tunnel perimeter, Einstein and Schwartz, 1979; Einstein and

Bobet, 1997; ITA, 2000 and Bobet, 2001. For the full drainage case the water pressure, u, at the

interface is zero and there is water flow towards the opening. For the no-drainage case the water

pressures are equal to far-field water pressure, uf, and there is no flow. Figure3 shows a tunnel with

and without drainage at the ground-liner interface.

For no-drainage conditions, the full water pressure is acting at the interface. The liner must deforminwards. Where, there is a compatibility of deformations between the ground and the liner at the

interface, the ground must follow the deformations of the liner and thus take some of the loading from

the water pressure. Hence, the water pressure at the interface is distributed between the ground and the

liner. The axial force is reduced due to the load transfer from the liner to the ground, which is a

function of their relative stiffness. The total stresses in the liner can be obtained directly using Eq. 3.

(3)

The problem of a tunnel with a liner and full drainage conditions at the liner-ground interface

can also be decomposed into the effective stress problem and the water pressure problem, Fig.

3. The tunnel is subjected to an inward internal pressure, uf, and full drainage at the tunnelwall, where it has been assumed that the tunnel has no liner. It will be shown that this

assumption is correct since the liner does not carry any water pressure, uf 0. Thus thestresses in the liner due to the water pressure are independent of the drainage conditions at the

contact between the liner and the ground, and can be obtained also from Eq. 3. This

conclusion is also reached by solving the same problem but using the concept of seepage

forces, Lambe and Whitman, 1969. The solution shows that indeed the stresses in the liner are

the same irrespective of the drainage conditions at the interface.

5. A calculation example

In order to verify the importance of a correct interpretation of the tunnel interaction for the conditionsof the ground water effect, 2D FE analyses are performed. The analyzed example concerns a circular

P

PF

eq

s

max

=

uu fhhfvv +=+= '' ,

xR

o

o

o

i


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tunnel with 8-m diameter and 0.4-m thickness of lining through homogeneous ground. Table 1

summarizes the conditions analyzed and the Geotechnical parameters. Modeling is performed using

Plaxis package based on Finite Element Method. The calculations are performed considering effective

soil stresses. A two-dimensional plane strain model is considered with a perfect elastoplastic Mohr-

Coulomb criterion for sandy soil and Cam-Clay criterion for clayey soil to describe the soil behaviorin the non-linear stage. 15-node isotropic linear strain triangular element was adopted for this analysis.

Figure 3: Decomposition of cases with drainage and no-drainage.

As the lining is not a continuous concrete ring, but is composed of 0.4-m thickness segments

connected via 0.24-m or more thickness section, the lining inertia is not constant. The stiffness at the

joint may be appreciably less than elsewhere. This can be accounted for through a decrease of stiffness

according to Muir Wood, 1975, which leads to a lesser stress within the lining. The segment joints are

never aligned along the tunnel and the thickness reduction is not as local as it is simulated in the

model, which is conservative.

Table 1. Material properties and parameters of soil.

Parameter Name Unit Fill Clay Sand

Levels m 0.0 3.0 3.0 7.0 7.0 35.0

Material model Model - Elastic CC MC

Material behavior Type - Drained Drained Drained

Dry soil weight dry KN/m3

15 16 18

Wet soil weight wet KN/m3

17 18 20

HllPermeability Kx m/day 1.0 e-5 1.0 e-7 1.0 e-2

VllPermeability Ky m/day 1.0 e-5 1.0 e-7 1.0 e-2

Youngs modulus E MPa 6 15 18

Liner increment ofE Eincr MPa - - 2

Reference level Yref m - - -9.0

Poissons ratio - 0.35 0.4 0.33

Cohesion C Kpa 10 40 1

Friction angle Degree 15.0 25.0 35.0

Dilatancy angle Degree 0.0 1.0 5.0

Normal consolidation slope * - - 0.25 -


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Elastic swelling line slope * - - 0.05 -

Lateral earth pressure at rest Ko - 1.0 0.577 0.45

Interface strength Rinter - Rigid Rigid 0.80

Interface permeability Perm - Neutral Neutral Neu./Ipmr.

/ Drain

The construction stage is applied to simulate the tunnelling process. The tunnel lining elements is

activated and the soil elements inside tunnel are deactivated. Deactivating the soil inside the tunnel

only affects the soil stiffness and strength and effective stresses. The water pressure inside the tunnel

also removed. The volume loss is simulated by applying a contraction to the tunnel lining. This

contraction is controlled by the deconfinement ratio .The vertical and horizontal effective stresses as well as the pore water pressure increase with depth.

The magnitude of the far-field stresses is obtained from the unit weight. The most important difference

between the previous analyses and this analysis is that the stresses along the lateral boundary increase

linearly with depth. The distance from the tunnel to the lateral and bottom boundaries of the area is

modeled with five to eight times the radius of tunnel.

Three cases are considered: first, without ground water (dry condition); second, drainage at theground-liner interface; third no-drainage conditions at the ground-liner interface. The ground water

level was assumed at 3.0-m below the ground surface for last two cases. Figure 4 shows the detail of

the mesh of the numerical model adopted to analyze the illustrated problem. The calculations were

performed in two stages to simulate TBM tunneling construction.

Figure 4: the developed mesh to study the illustrated examples.

6. Results of calculations

Figure 5 shows the total deformation of the mesh for the three model cases and Fig. 6 illustrates the

arrows lines of the displacements in the soil mass for the same cases. The surface settlement for the

three models is illustrated in Fig. 6. The comparisons between the calculation results of the vertical

settlements are given in Table 2. The load displacement curve at the surface point of centerline for the

models is shown in Fig. 7 at the second stage of calculation.

The results of the internal forces are developed in the tunnel lining according to the construction stages

and the conditions of every model case. The bending moments, the shearing forces and the normal

forces are compared for the three-model case in Fig. 8. The comparisons between the maximumresults of the internal forces are given in Table 2. As a result, the normal force is less after the first


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stage of calculation. The bending moments, however, are larger. The deformation of lining for the

three model cases is illustrated in Fig. 9.

The contours of the vertical effective stress at the final loading for the three models are shown in Fig.

10. The plastic regions show that the soil failure occurred in a limited area around the tunnel during itsconstruction as shown in Fig. 11 for the three models.

The number of iteration steps for convergence in calculations for the three models are given in Table 2

for comparison.

Table 2. Results of the analysis for comparison.

Model type

Descriptions

Model (1)

Without ground water

Model (2)

Drainage

Model (3)

No-Drainage

Surface settlements 27.5 mm 32.0 mm 22.0 mm

Tunnel deformations 48.0 mm 52.5 mm 38.0 mm

Normal forces -340.2 KN/m -202.2 KN/m -667.7 KN/m

Shearing forces -53.3 KN/m -48.8 KN/m -30.1 KN/mBending moments 96.3 KNm/m 87.3 KNm/m 56.5 KNm/m

Iteration steps 8 65 43

model (1)

model (2)


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model (3)

Figure 5: Total deformation of the mesh

model (1)

model (2)


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model (3)

Figure 6: Total displacements of the mesh

0 5.00E-03 0.010 0.015 0.020 0.025 0.030

0.0

0.3

0.6

0.9

1.2

|U| m

Sum-McontrA

model (1)


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0 5.00E-03 0.010 0.015 0.020 0.025 0.030 0.035

0.0

0.3

0.6

0.9

1.2

|U| [m]

Sum-McontrA

model (2)

0 5.00E-03 0.010 0.015 0.020 0.025

0.0

0.3

0.6

0.9

1.2

|U| [m]

Sum-McontrA

model (3)

Figure 7: Load Displacement curve at surface point

Normal forces Shearing forces Bending moment

Model (1)


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Model (2)


Model (3)

Figure 8: Comparison for internal forces for the three model cases.

Total displacementsExtreme total displacement 48.87*10-3 m Total displacementsExtreme totaldisplacement 52.56*10-3 m

Total displacementsExtreme total displacement 37.89*10

-3m

Model (1) Model (2) Model (3)

Figure 9: Comparison of the deformation in lining for the three model cases.


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Model (1)

Model (2)

Model (3)


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Figure 10: Comparison of the effective stresses for the three model cases.

Plastic Points

Plastic Mohr-Coulomb point

Plastic cap point Tension cut-off point Model (1)

Plastic Points


Plastic c ap point Tensi on cut-off poi nt Model (2)

Plastic Points


Plastic cap point Tension cut-off point


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Model (3)

Figure 11: Comparison of the plastic zones around the tunnel for the three model cases.

7. Conclusions

The interaction of support system with soil in TBM tunnelling has been studied in this work using the

convergence-confinement method. The effects of drainage and no-drainage conditions at the ground

liner interface are investigated and compared with the case of no ground water (dry condition). It is

observed that the ground displacements and total stresses at the ground liner interface depend on the

drainage conditions. The analysis shows that the current practice of applying the full water pressure to

an impermeable liner is acceptably and conservative while for liner drainage it is quite significant. So,

the choosing of the materials that are used to close the tail gap of TBM processing is an important

issue. Also, the water-tightness material of the liner segments is playing an important role to control

the drainage of ground water through liner.

The existence of groundwater condition may seriously affect the internal forces of liner and its

deformations. Also, it affects the soil mass stresses and strength and its deformations. The relationship

between the effective stress and the groundwater pressure is inversely proportional. The consolidationanalysis however may be more reasonable for the groundwater condition.

8. References

A.F.T.E.S.,1993.Groupe de travail n. 7-Souttenement et revetement, Emploi de la mthode

convergence-cofinement, Tunnels et ouvrages souterrains, Supplment au n. 117, Maj-Juin, pp. 118-

205.

Peila, D., Orests, P.P., 1995. Axisymmetrical analysis of ground reinforcing in tunnelling design,

computer and Geotechaics, Vol. 17. Elsevier Science Ltd, London, UK, pp. 235-274.

Panet M., 1995. Le calcul des tunnels par la mthode convergence-cofinement, Presses de Lcole

Nationale des Ponts et Chausses, Paris.

Panet, M., and Guent, A., 1982. Analysis of convergence behind the face of a tunnel. Proc. Tunnelling,

Brighton 82, 197-204.

Bobet, A., 2001. Analytical solutions for shallow tunnels in saturated ground. ASCE J. Eng. Mech. 12,

1258-1266.

Einstein, H.H., and Bobet, A., 1997. Mechanized tunnelling in squeezing ground-from basic thoughts

to continuous tunnelling. Proceedings of the world tunnel congress97. Vienna, Austria, pp. 619-632.

Einstein, H.H., and Schwartz, C.W., 1979. Simplified analysis for tunnel supports. J. Geotech. Eng.

Division, ASCE 105, 499-518.

International Tunnelling Association (ITA), Working Group No. 2, 2000. Guidelines for the design of

shield Tunnel Lining, Tunnelling and Underground Space Technology, 15 (3), pp. 303-331.

Lambe, T.W., Whitman, R.V., 1969. Soil Mechanics. Wiley, New York.

Muir Wood, A.M., 1975. The Circular Tunnel in Elastic Ground, Geotechnique 25, No. 1, pp 115-

127.

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