Design of Lining of Tunnels Excavated in Soil and Soft Rock.pdf

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Design of Lining of Tunnels

Excavated in Soil and Soft Rock

Massoud Palassi

Department of Civil Engineering, University of Tehran, Iran

[email protected]

Mohammad M. Mohebbi

Lar Consulting Engineers, Tehran, Iran

[email protected]

ABSTRACT

Most of the tunnels constructed in urban areas include excavation in soil or soft rock. In this

paper, a comparison is made between a numerical method based on the finite element method

(FEM) and two analytical methods in estimating the bending moment and axial forcesdeveloped in the lining of these tunnels. The analytical methods employed in this article

include Curtiss and Muir-Woods methods. It is shown that there is a good compatibility

between the numerical modelling and these analytical methods. The effects of various

parameters on the results are also evaluated.

KEYWORDS: tunnel; lining; analytical method; numerical method

INTRODUCTION

Construction of tunnels in urban areas, for various purposes such as underground

transportation and sewage disposal, usually includes excavation in soil or semi-rigid media.

Unlike hard and massive rocks, these types of ground normally need to be supported immediatelyafter excavation in order to avoid collapse and to constrain the surface settlements which can be

critical in the urban areas. New methods of tunneling such as pipe jacking and mechanizedtunneling by tunnel boring machines (TBMs) have made it possible to excavate tunnels in soft

ground and install the support system simultaneously. Use of these methods combined withproper lining design results in better rates of advance, higher stability and lower surfacesettlements.


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Rapid progresses in the development of user friendly computer codes and limitations of theanalytical methods have increased the tendency of designers to use numerical methods for the

design of tunnel linings. On the other hand, the analytical and analytical-empirical methods are

based on valuable experience and have been successfully used for the design of tunnels in thepast. In this paper, a comparison is made between a finite element based numerical method usinga commercially available program and two analytical methods for estimating the bendingmoments and axial forces developed in the lining of tunnels. The effects of different parameters

on the results are also evaluated.

ANALYTICAL METHODS

Muir-Woods Method

Muir-Wood (1975) presented a simple approach to the problem of proportioning the tunnel

linings which can be useful in many situations. He assumes that a circular lining deforms into anelliptical shape in an elastic ground. It is also assumed that a full interaction between lining andthe surrounding ground exists. The Airy stress function is used in the polar coordinates(according to the stress function proposed by Morgan):

dcrbrar 2cos)242

( +++= (1)

which allows the differential equations of equilibrium to take the form

2

2

2

11

rr

rr

+

= (2)

2

2

r

= (3)

=

rrr

1(4)

in which

r : radial stress in the ground,

: tangential stress in the ground, and

r

: shear stress in the ground.

Using the plane strain condition and assuming that no shear stress exists between the liningand the surrounding soil, the maximum bending moment and axial force developed in the lining

can be obtained from:


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sR1

sR

2eRd

p3

1maxM

+= (5)

cR1

1upeRmaxN

+= (6)

in which

2

hpvp

dp

= (7)

eR

R = (8)

4eR

3

lIl9E

sR = (9)

2

hpvp

up+

= (10)

)gt(1lE

)2l(1gEeR

cR+

= (11)

vkphp = (12)

c)Rg6)(5g(1

g3E

+= (13)

vp : vertical pressure of the ground

hp : horizontal pressure of the ground

eR : external radius of the lining

R : middle radius of the lining

k : coefficient of lateral earth pressurelE : elastic modulus of the lining

lI : moment of inertia of the lining

lA : cross sectional area of the lining

l : Poissons ratio of the lining

gE : elastic modulus of the ground


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g : Poissons ratio of the ground

t: lining thickness

Curtiss Method

In the method proposed by Curtis (1976) the bending moments and axial forces developed inthe lining of the tunnel are estimated similar to the Muir-Woods method except that the shear

stress between the lining and the ground is taken into the account. The maximum bendingmoment and the maximum axial force developed in the lining is calculated as:

)tlSnl(2S6

eRR

maxM += (14)

)tlSnl(S3

eR

1Q1

1eRupmaxN +

+= (15)

in which

g43

g23

2Q1

2Q1

dpnlS

+

= (16)

nlS

2Q1

22Q1

tlS

+= (17)

l)Ag(1lE

)l)R(1l(1gE

1Q+

+= (18)

)g(1lIl12E

g)E2l(1

3R

2Q+

= (19)

lA,lI,lE,l,gE,g,dp,up,eRR, are defined the same as in Muir-Woods method.

Estimation of the Ground Pressure

One approach to estimate the ground pressure acting on the lining is to apply the fulloverburden. Another approach based on I.T.A (2000) recommendation is to reduce the

overburden according to the Terzaghis formula:

+

+

= R

8

2tan

HtanKe1

2K.tan

C))8

2(R.cot(

vP (20)

in which


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vp : vertical pressure of the ground

R : radius of the lining : internal friction angle of the soil

C : cohesion of the soilK : coefficient of lateral earth pressure

H : depth of tunnel

In this paper to evaluate the ground pressure in Curtiss method both of the above mentioned

approaches have been used.

NUMERICAL ANALYSIS

For the numerical analysis the Finite Element Method (FEM) is used involving 2D plane

strain conditions and a suitable modeling of soil nonlinearity by step-by-step construction

simulation. In this study, a commercially available finite element software, PLAXIS, has beenemployed. The first step of modeling is the creation of the geometry which includes layers of thesoil and the tunnel cross section. To avoid the boundary interference on the results, boundarieshave been taken at least 3 times the tunnel diameter from the external surface of the lining. Then,

boundary conditions have been applied. Finally, properties of the soil and its constitutive law andthe lining properties have been included.

The mesh is automatically generated based on the triangulation procedure. Once the 2DFinite Element mesh has been generated, calculations can be performed. It is also necessary todefine the construction stages. In this paper, it is assumed that the tunnel excavation and lininginstallation take place simultaneously.

COMPARISON BETWEEN NUMERICAL ANDANALYTICAL METHODS

According to Muir-Woods and Curtis' formulae, the parameters affecting the bending

moments and the axial forces developed in the lining are tunnel depth, tunnel radius, coefficientof lateral earth pressure, elastic modulus and Poissons ratio of the ground and elastic modulusand thickness of the lining. Depending on the approach used for the evaluation of the ground

pressure, the internal friction angle and the cohesion of soil can also influence the bendingmoment and the axial force. Soil and lining properties used for the numerical and analytical

solutions are presented in Table 1.

In the following, the effects of the above-mentioned parameters are evaluated. It should benoted that Curtis (FL) means Curtiss method using the approach of applying full load of theoverburden and Curtis (PL) means Curtiss method involving the approach of applying partialload of the overburden according to the Terzaghis relationship.


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Tunnel Depth

Figures 1 and 2 show the effect of the tunnel depth. As shown in Figure 1, the results

obtained from Curtis (FL) method are close to the results obtained from FEM. Figure 2 showsthat in the analytical methods, the maximum axial forces are overestimated in comparison with

FEM results except in Curtis (PL) method in high depths.

Table 1: Ground and lining properties

ValueUnitParameter

30mTunnel depth

5mTunnel radius

0.5-Coefficient of lateral earth pressure

100MPaElastic modulus of soil

0.3-Soil Poissons ratio

30kPaSoil cohesion30Soil internal friction angle

18kN/mSoil unit weight

30GPaLining elastic modulus

0.25mLining thickness

24kN/mLining unit weight

0.15-Lining Poissons ratio

Figure 1: Maximum bending moment vs. the depth of tunnel


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Figure 2: Maximum axial force vs. the depth of tunnel

Tunnel Radius

The effects of the tunnel radius are illustrated in Figures 3 and 4. Figure 3 shows that themaximum bending moments decrease with the increase of the tunnel radius. This can beexplained by noting the fact that an increase in the tunnel radius leads to a decrease in the bending

stiffness of the lining. Therefore, the lining acts as a membrane in which significant bendingmoments cannot develop. In the same manner, the increase of the axial forces can be explained bymembrane behaviour of the lining. As shown in Figures 3 and 4, the results obtained from thenumerical method for the maximum bending moment and the axial force are more close to theresults given by Curtis (FL) method and Curtis (PL) method, respectively. According to Figure 4,

the relation between the maximum bending moments the radius of tunnel is linear.

Figure 3: Maximum bending moment vs. the radius of tunnel


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Figure 4: Maximum axial force vs. the radius of tunnel

Coefficient of Lateral Earth Pressure

Figures 5 and 6 show the effect of the lateral earth pressure. In all of the solutions, the

maximum bending moment developed in the lining decreases rapidly with the increase in thecoefficient of the lateral earth pressure because a relatively uniform compressive stress developsin the lining. Figure 5 shows that the results obtained for the maximum bending moment fromCurtis (FL) method is closer to the results of FEM. According to Figure 6, the results obtained forthe maximum axial force from Curtis (PL) are almost the same as the results obtained from FEM.

Figure 5: Maximum bending moment vs. the lateral earth pressurecoefficient


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Figure 6: Maximum axial force vs. the lateral earth pressure coefficient

Elastic Modulus of the Soil

Figures 7 and 8 show effect of the elastic modulus of soil. It can be seen that the maximum

bending moments calculated from both FEM and analytical methods decrease with the increase ofthe elastic modulus of the soil. This is reasonable because if the elastic modulus of the soil

approaches to the elastic modulus of the lining, no bending moment develops in the lining. AlsoFigures 7 and 8 show the compatibility of Curtis (FL) and Curtis (PL) methods with FEMsolution for estimation of the maximum bending moment and the maximum axial force,

respectively.

Figure 7: Maximum bending moment vs. the elastic modulus of soil.


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Figure 8: Maximum axial force vs. the elastic modulus of soil

Poissons Ratio of the Soil

Effects of the Poissons ratio of the soil are given in Figures 9 and 10. Figure 9 shows thatwith the increase of Poissons ratio of the soil there are small amounts of decrease in themaximum bending moment. This Figure also shows a rather good compatibility between FEM

solution and Curtis (FL) method. According to Figure 10, the maximum axial force is notsensitive to the Poissons ratio of the soil. Also in this Figure, a very good compatibility can beseen between FEM solution and Curtis (PL) method.

Figure 9: Maximum bending moment vs. the Poissons ratio of soil


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Figure 10: Maximum axial force vs. the Poissons ratio of soil

Cohesion of the Soil

Muir-Woods and Curtiss formulae show that the cohesion of the soil has not any effect onthe maximum bending moment and the maximum axial force except for the case of decreasingthe overburden by Terzaghis formula. Figure 11 shows that the cohesion of the soil does not

affect the maximum bending moment calculated by FEM. This Figure also shows a relativelygood compatibility between FEM solution and Curtis (FL) method. A good compatibility can benoted between FEM solution and Curtis (PL) method for estimating the maximum axial force inFigure 12.

Figure 11: Maximum bending moment vs. the cohesion of soil


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Figure 12: Maximum axial force vs. the cohesion of soil

Internal Friction Angle of the Soil

The internal friction angle of the soil does not affect the maximum bending moment and the

maximum axial force calculated by Curtiss and Muir-Woods methods except for the case ofdecreasing the overburden by Terzaghis formula. But the internal friction angle affects both the

bending moment and the axial force calculated by FEM (Figures 13 and 14). Figures 13 and 14also show that Curtis (FL) and Curtis (PL) methods estimate the maximum bending moment andthe maximum axial force more closely to the FEM solution, respectively.

Figure 13: Maximum bending moment vs. the internal friction angle of soil


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Figure 14: Maximum axial force vs. the internal friction angle of soil

Elastic Modulus of the Lining

The effects of the elastic modulus of the lining are illustrated in Figures 15 and 16. In allsolutions, an increase in the elastic modulus of the lining leads to an increase in the liningstiffness and so an increase in the lining forces. Figures 15 shows a rather good compatibility

between Curtis (FL) with FEM for maximum bending moment and Figure 16 shows a goodcompatibility between Curtis (PL) method with FEM solution for the maximum axial force.

Figure 15: Maximum bending moment vs. the elastic modulus of the lining


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Figure 16: Maximum axial force vs. the elastic modulus of lining

Thickness of the Lining

The effects of the lining thickness are shown in Figures 17 and 18. The lining thickness has asignificant effect on the maximum bending moment. It can be explained by noting the fact that anincrease in the thickness results in an increase in the bending stiffness of the lining. A goodcompatibility exists between Curtis (FL) method and FEM solution for the estimation of the

maximum bending moment. Curtis (PL) method and FEM solution give the same results for themaximum axial force.

Figure 17: Maximum bending moment vs. the lining thickness


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Figure 18: Maximum axial force vs. the lining thickness

CONCLUSIONSA comparison was made between a numerical analysis using a geotechnical finite element

method (FEM) program and two analytical methods, namely Curtiss and Muir-Woods methods,for the estimation of the maximum bending moment and maximum axial force developed in thelining of tunnels. It was shown that there is a rather good compatibility between the results of theFEM and Curtis (FL) method for the estimation of the maximum bending moment developed inthe lining except that Curtis (FL) results are often less than FEM results. There is good

compatibility between FEM results and Curtis (PL) method for the estimation of the maximumaxial force. The maximum bending moment developed in the lining is almost insensitive to thePoissons ratio and the cohesion of the soil, but it is very sensitive to the lining thickness. Themaximum axial force is rather insensitive to the Poissons ratio of the soil and the elastic modulus

of the lining.

REFERENCES1. Brinkgreve, R.B.J. and P.A. Vermeer (1998) PLAXIS Manual, BALKEMA.

2. Curtis, D.J. (1976) Discussion on the Circular Tunnel in Elastic Ground, Geotechnique, Vol. 26,No. 1, pp. 231-237.

3. International Tunneling Association (ITA), Working Group No.2 (2000) Guidelines for the Design

of Shield Tunnel Lining, Tunnelling and Underground Space Technology, Vol. 15, pp. 303-331.

4. Muir-Wood, A.M. (1975) The Circular Tunnel in Elastic Ground, Geotechniqe, Vol. 25, N0.1, pp.

115-127.

2008 ejge
http://www.ejge.com/copynote.htmhttp://www.ejge.com/copynote.htm

Design of Lining of Tunnels Excavated in Soil and Soft Rock.pdf

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