Analytic stress solutions for a circular pressure tunnel at pressure and great depth including support delay

International Journal of Rock Mechanics & Mining Sciences 48 (2011) 514–519

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Technical Note

Analytic stress solutions for a circular pressure tunnel at pressure and great depth including support delay Ai-zhong Lu a,, Lu-qing Zhang b, Ning Zhang a a b

Institute of Hydroelectric and Geotechnical Engineering, North China Electric Power University, Beijing 102206, China Key Laboratory of Engineering Geomechanics, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China

a r t i c l e i n f o Article history: Received 3 June 2010 Received in revised form 6 September 2010 Accepted 10 September 2010 Available online 19 November 2010

1. Introduction Underground tunnels are widely used in hydropower, traffic, mining and military engineering to ensure the tunnel safety, concrete lining is applied in the tunnel. A number of methods have been proposed to calculate the stress and displacement in the surrounding rock mass and the lining, one of which is the analytical method. For instance, the complex function method developed by Muskhelishvili [1] is especially suitable for solving underground tunnel problems [2]. However, this method has limited applications, which mainly are on plane elastic problems. Numerical methods are an alternative way. For instance, the finite element method has been widely used in underground engineering. This method can be used to solve the elasto-plastic problems, which can hardly be solved by the complex function method. Especially, when the structure is composed of multiple materials, and the stress and displacement boundary conditions are complex, the analytical method can hardly find the solution, while the numerical method is still effective. However, the closed-form solution obtained by the analytical method is valuable and helpful for in-depth analysis of the problem [3]. The closed-form solutions for unsupported tunnel at great depth have been proposed for single circular tunnel [4,5] and tunnels with complex geometry, such as elliptic tunnel, rectangular tunnel, semicircular tunnel, inverted U-shaped tunnel and notched circular openings [2,6–9]. The complex function method can be applied to find the stress solutions not only for single tunnel in arbitrary shape, but also for multiple tunnels with arbitrary shape [10–13]. As for elastic solutions for deep supported tunnels, the plane strain problem associated with a single circular tunnel with ring lining in an infinite domain has been studied in depth [14,15].

 Corresponding author. Tel.: +86 010 61772392.

E-mail addresses: [email protected], [email protected] (A.-z. Lu). 1365-1609/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2010.09.002

Theoretically, the solutions based on plain strain can only be applicable to the situation that lining is applied immediately after excavation and no deformation occurs in the surrounding rock mass before that. In fact, after the tunnel is excavated by a certain length and before the lining is applied, some deformation has occurred in the surrounding rock mass. After lining installation, as the working face is advanced, the surrounding rock mass experiences further deformation and leads to forces on the lining. Although the support delay process was considered in Refs. [14,15], some limitations still exist. After the tunnel is excavated, some instantaneous elastic deformation occurs in the surrounding rock mass; however, the deformation is not finished. The amount of deformation completed is related to the distance between the working face and the supported section. This is due to the spatial constraint effect exerted by the unexcavated rock mass in front of the working face. Within a certain distance from the working face, the displacement in the surrounding rock mass varied with the distance [16,17]. Li and Liu [16] defined the ratio between this part of displacement and the maximum displacement at any cross section between the working face and unsupported section as the displacement release coefficient b. The coefficients for various forms of cross sections and various distances to the working face for unsupported tunnel were obtained by three-dimensional finite element method. The displacement release coefficient at the working face was in the range 0.20–0.27. When the distance to the working face was 2.5D, where D is the larger value between the height and width of the tunnel cross section, the release coefficient approached 1. It means that the larger the distance between the working face and the supported section, the more the deformation completed in the surrounding rock mass. If the lining is applied at this moment, the stress on lining is smaller. Excavation and support of underground tunnels is a spatial problem. If it is assumed to be a plane strain problem, the displacement completed before liner application can be obtained by the methods proposed in [14,15]. Bulychev [14] introduced a

Ai-zhong Lu et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 514–519

p

515

p

y

y Xn

P0

x

λp

N Yn R1

x

λp

R0 R1

Fig. 1. Lined circular tunnel under uniform internal pressure and an initial stress field.

coefficient a* to reflect the excavation process with support delay. Before the stress and displacement were solved, a* was unknown, instead, it was back-calculated through the stress distribution in the lining by field measurement. In Ref. [14], the tunnel was only subjected to in situ stresses and no hydrostatic pressure in the lining. In Ref. [15], a coefficient Z was introduced to reflect the displacement completed before support installation. However, only the radial displacement was considered in the formulas and the tangential displacement was neglected. Only when the principal in situ stresses are equal in the horizontal and vertical directions for a horizontal tunnel and uniformly distributed, the problem under discussion is axisymmetrical and no tangential displacement occurs. In the case study presented in Ref. [15], the displacement completed before support installation was not considered. This means that the tunnel was supported immediately after excavation and before any deformation took place in the surrounding rock mass. The more general plane strain problem of a circular hole, with two linings of different materials has been solved by Atkinson and Eftaxiopoulos [18,19], for pure bond and pure slip boundary conditions not including support delay. Both the in-plane and the anti-plane problems have been considered in [18,19]. In this study, the displacement release coefficient is determined according to the distance between the working face and the supported section. In other words, given the displacement completed in the surrounding rock mass before support installation, the analytic stress solutions for a circular pressure tunnel at great depth (Fig. 1) can be obtained. Similar to Refs. [14,15], the following assumptions are made: (1) the surrounding rock mass and the lining are always linear elastic under the action of in situ stresses and hydrostatic pressure and (2) the tunnel is deep enough, so that the problem can be simplified to be an infinite domain problem.

2. Fundamental theories for solution 2.1. Total displacement induced by excavation without support Before tunnel excavation, the rock mass is in initial stress state s0x ¼ lp, s0y ¼ p, s0xy ¼ 0 (in Fig. 2, l is the lateral pressure coefficient; sign convention is defined as positive for tension and negative for compression; p is the vertical component of in situ stress and it is positive.) The surface forces at the boundary are released by excavation, which leads to displacement in the

Fig. 2. Surface force components on the boundary of an unsupported circular tunnel.

surrounding rock mass. Let {Xn, Yn} denote the components of surface force on the boundary in the x and y directions, respectively. They can be determined by the initial stress field before excavation and described as follows [2]: Xn þ iYn ¼ lp

dy dx þ ip ds ds

ð1Þ

where ds is arc length, and its positive sense is clockwise direction. Tunnel excavation can be understood as application of a surface with magnitude of  (Xn + iYn) on the working face so that the resultant surface force is zero. From Eq. (1), we have I pð1 þ lÞ pð1lÞ zþ z, z ¼ t ¼ R1 eiy ð2Þ i ðXn þ iYn Þ ds ¼ 2 2 Eq. (2) can be integrated in the clockwise direction. If the tunnel is unsupported, the stress boundary condition at the tunnel boundary (r ¼R1) can be written as [1,2]

jðtÞ þt ju ðtÞ þ cðtÞ ¼

pð1 þ lÞ pð1lÞ tþ t 2 2

ð3Þ

where

jðtÞ ¼

1 X

ak t k ,

cðtÞ ¼

k¼1

1 X

bk t k

ð4Þ

k¼1

The functions j(z) and c(z) in Eq. (4) are two analytic functions in the domain defined by rZR1. As the tunnel geometry and external loads are symmetrical about the x and y axes, both ak and bk are real numbers. According to Cauchy integral method or power series solution, j(z) and c(z) can be determined by substituting Eq. (4) into Eq. (3), which can be written as

jðzÞ ¼

pð1lÞ 2 1 R1 , 2 z

cðzÞ ¼

pð1 þ lÞ 2 1 pð1lÞ 4 1 R1 þ R1 3 2 z 2 z

ð5Þ

Substituting Eq. (5) into Eq. (6) [1,2] yields ur þ iuy ¼

1 iy e ½k1 jðzÞzju ðzÞcðzÞ 2G1

ð6Þ

The radial displacement ur and the tangential displacement uy at any point in the surrounding rock mass can be obtained as follows:  1 iy pð1lÞ 2 1 pð1lÞ 2 z R1 þ R1 2 e k1 ur þ iuy ¼ 2G1 2 z 2 z  pð1 þ lÞ 2 1 pð1lÞ 4 1 R1  R1 3 ð7Þ  2 2 z z

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Ai-zhong Lu et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 514–519

where k1 ¼3–4m1, m1 and G1 are Poisson’s ratio and shear modulus of the rock, respectively, G1 ¼E1/2(1+ m1), and E1 is Young’s modulus of the rock. The displacement shown in Eq. (7) is the total displacement in the surrounding rock mass when the tunnel is unsupported.

Substituting Eq. (10) into Eq. (13), we have 3 3 2f1 þ g1 R2 0 þ 2ðf2 R0 þ g2 R0 Þcos y þ 2iðf2 R0 þ g2 R0 Þsin y 1 X

þ

½2kek Rk1 þ 2ðk þ 2Þfk þ 2 Rk0 þ 1 þ ðk þ 2Þgk þ 2 Rk3 cosðk þ 1Þy 0 0

k¼1

2.2. Problem-solving principles of two complex potential functions considering support installation process

¼ Zður þiuy Þ

½kðk þ 1Þek Rk1 ðk þ 2Þðk þ 1Þfk þ 2 Rk0 þ 1 khk Rk1 0 0 cosðk þ 1Þy

k¼1

If the support is installed when the displacement is Z times of the total displacement, where 0 r Z r1, the displacement occurs before support installation can be written as ur þ iuy

1 X

þ

þi

1 X

½ðk þ 2Þgk þ 2 Rk3 þ kðk þ 1Þek Rk1 0 0

k¼1

ðk þ 2Þðk þ 1Þfk þ 2 Rk0 þ 1 khk Rk1 0 sinðk þ 1Þy ¼ p0

ð14Þ

ð8Þ Eq. (14) is applicable to arbitrary y. Hence

In fact, Z is the displacement release coefficient b defined in [16], which is determined by the distance between the working face and the lined section. If Z ¼0, the tunnel is supported immediately after excavation and no deformation occurs in the surrounding rock mass before support installation. This is not the real case in engineering practices. It is known from Ref. [16], that 20% of the total displacement has taken place at the working face after excavation. Therefore, some deformation inevitably occurs in the surrounding rock mass even if the support is installed immediately after excavation. If Z ¼1, the support is installed when the total displacement is completed. After the lining is applied, the surrounding rock mass and the lining interact with each other. The two analytic functions corresponding to the surrounding rock mass under the action of lining can be expressed by Taylor series [1,2]

j1 ðzÞ ¼

1 X

ck zk ,

c1 ðzÞ ¼

k¼1

1 X

dk zk

ð9Þ

k¼1

On the other hand, the two complex potential functions corresponding to the lining under the action of the surrounding rock mass can be expressed by Laurent series [1,2]

j2 ðzÞ ¼

1 X

ek z

k

þ

k¼1

1 X

k

fk z ,

c2 ðzÞ ¼

k¼1

1 X

gk z

k

k¼1

þ

k

hk z

j1 ðzÞ þzj

ður þ iuy Þður þ iuy Þ þ ¼

iy

¼ j2 ðzÞ þzj

z ¼ R1 e

u 2 ðzÞ þ c2 ðzÞ,

1 iy e ½k2 j2 ðzÞzj2u ðzÞc2 ðzÞ, 2G2

z ¼ R1 eiy

ð16Þ

kek Rk1 þfk þ 2 Rk0 þ 1 þgk þ 2 Rk3 ¼ 0, 0 0

k Z1

ð17Þ

By substituting Eqs. (7)–(11) into Eq. (12), the following equations can be obtained: 1 X ð1ZÞp 1 iky ½k1 ð1lÞR1 eiy ð1 þ lÞR1 eiy  þ ðk1 þ G1 =G2 Þ ck Rk 1 e 2G1 G1 k¼1 " # 1 1 X ðG1 =G2 1Þ X k þ 2 iky k iky  ðk2Þck2 R1 e  dk R1 e G1 k¼3 " #k ¼ 1 1 1 X ðk2 þ 1Þ X iky ek Rk þ fk Rk1 eiky ð18Þ ¼ 1 e G2 k¼1 k¼1

By comparing the coefficients of eiky in Eq. (18), we have ð1ZÞp 1 1 k1 ð1lÞR1 þ ðk1 þ G1 =G2 ÞR1 ðk2 þ1ÞR1 1 c1 ¼ 1 e1 2G1 G1 G2

ð19Þ

ð1ZÞp 1 1 ð1 þ lÞR1 þ ðG1 =G2 1ÞR1 ðk2 þ 1ÞR1 f1 1 d1 ¼ 2G1 G1 G2

ð20Þ

1 1 ðk1 þ G1 =G2 Þck ¼ ðk2 þ 1Þek , G1 G2



ð21Þ ð22Þ

1 1 ðk2 þ1Þ ðG1 =G2 1Þck Rk ðG1 =G2 1Þdk þ 2 Rk2 ¼ fk þ 2 R1k þ 2 , 1 þ 1 G1 G1 G2

Substitution of Eqs. (9) and (10) into Eq. (11) leads to 1 X

iky ck Rk  1 e

k¼1

¼

1 X

ðk2Þck2 R1k þ 2 eiky þ

k¼3 1 X

iky dk Rk 1 e

k¼1 1 X

iky ek Rk þ 1 e

k¼1

1 X

1 X

fk Rk1 eiky 

k¼1 1 X

þ f1 R1 eiy þ 2f2 R21 þ

þ 2 iky ðk2Þek2 Rk e 1

k¼3

ðkþ 2Þfk þ 2 Rk1 þ 2 eiky

k¼1 1 X

iky gk Rk þ 1 e

k¼1

1 X

hk Rk1 eiky

ð24Þ

k¼1

By comparing the coefficients of eiky in Eq. (24), we have 1 d1 R1 1 ¼ 2f1 R1 þ g1 R1

u 2iy ½z 2 ðzÞe

2Re ½j

j

00

u 2 ðzÞ þ c2 ðzÞ ¼

p0 ,

iy

z ¼ R0 e

ð13Þ

where k2 ¼3–4m2, m2 and G2 are Poisson’s ratio and shear modulus of the lining, respectively, G2 ¼E2/2(1+ m2), and E2 is Young’s modulus of the lining.

kZ1

ð23Þ

þ

ð12Þ

kZ 2

d2 ¼ 0

ð11Þ

1 iy e ½k1 j1 ðzÞzj1u ðzÞc1 ðzÞ 2G1

ð15Þ k Z1

ð10Þ

k¼1

g2 ¼ f2 ¼ 0

þðk þ2Þfk þ 2 R0k þ 1 þhk Rk1 ¼ 0, ek Rk1 0 0



1 X

where ck, dk, ek and fk in Eqs. (9) and (10) are real numbers to be determined by the boundary conditions given below. Suppose that a uniform hydrostatic pressure p0 acts on the inner boundary of the lining. As the lining and the surrounding rock mass interact with each other, the two complex potential functions corresponding to the surrounding rock mass and the lining under the combined action of in situ stresses and hydrostatic pressure can be still in the form of Eqs. (9) and (10). It is assumed that the lining is in full contact with the surrounding rock mass. Therefore, by the stress and displacement continuity conditions at the interface between the surrounding rock mass (r ¼R1) and the lining and the stress boundary condition (Fig. 1) at the inner boundary of the lining (r ¼R0), the following equations can obtained: u 1 ðzÞ þ c1 ðzÞ

2f1 þ g1 R2 0 ¼ p0 ,

k kþ2 ck Rk þ hk Rk1 , 1 ¼ ek R1 þ ðkþ 2Þfk þ 2 R1

ð25Þ k Z1

k2 k2 ¼ fk þ 2 Rk1 þ 2 kek Rk , kck Rk 1 þdk þ 2 R1 1 þ gk þ 2 R1

ð26Þ kZ1

ð27Þ

Ai-zhong Lu et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 514–519

It can be seen from Eqs. (16), (17), (21), (23), (26) and (27) that, when kZ2, the coefficients ck, dk + 2, ek, fk + 2, gk + 2 and hk are zero and c1, d1, d3, e1, f1, f3, g1, g3 and h1 are non-zero. These coefficients can be determined by the linear Eqs. (28)–(36) composed by Eqs. (15)–(17), (19), (20), (23), (25)–(27) 2R20 f1 þ g1 ¼ p0 R20

ð28Þ

e1 þ 3R40 f3 þR20 h1 ¼ 0

ð29Þ

R20 e1 R60 f3 g3 ¼ 0

ð30Þ

ðG1 =G2 1Þd1 

where q1 ¼ 2f1 þ g1 R1 2 , 2

q3 ¼ 6f3 R1 2e1 R1

G1 ð1ZÞp ð1þ lÞR21 ðk2 þ1ÞR21 f1 ¼ 2 G2

ð31Þ

c1 ðzÞ ¼  ð32Þ

q2 ¼ 4e1 R1 2 þ3g3 R1 4 h1 and 2

þ3g3 R1 4 þ h1

ð45Þ

The total stress in the surrounding rock mass is the superposition of the stress before excavation, the stress after excavation and the stress after support installation. Correspondingly, the two complex potential functions are the superimposition of Eqs. (5), (37) and the complex potential functions before excavation, i.e.,

j1 ðzÞ ¼ 

G1 ð1ZÞp ðk2 þ 1Þe1 ¼  k1 ð1lÞR21 ðk1 þG1 =G2 Þc1  2 G2

517

pð1 þ lÞ pð1lÞ 2 1 zþ R1 z þ c1 z1 4 2

ð46Þ

pð1lÞ pð1 þ lÞ 2 1 pð1lÞ 4 3 zþ R1 z þ R1 z þ d1 z1 þ d3 z3 2 2 2 ð47Þ

ð33Þ

The first term in Eqs. (46) and (47) represents the corresponding complex potential function before excavation. The stresses in the surrounding rock mass sr1, sy1 and try1, can be obtained by replacing j2(z) and c2(z) in Eqs. (39) and (40) by j1(z) and c1(z)

d1 2R21 f1 g1 ¼ 0

ð34Þ

sr1 ¼ ð1 þ lÞp=2 þ ½ð1 þ lÞpR21 =2 þ d1 =r 2 þfð1lÞp=2

c1 e1 3R41 f3 R21 h1 ¼ 0

ð35Þ

R21 c1 d3 R21 e1 þ R61 f3 þ g3 ¼ 0

ð36Þ

ðG1 =G2 1ÞR21 c1 ðG1 =G2 1Þd3 þ

G1 ðk2 þ 1ÞR61 f3 ¼ 0 G2

Hence, the solutions by the complex potential functions for the surrounding rock mass and the lining are obtained.

2½ð1lÞpR21 þ2c1 =r 2 þ ½3ð1lÞpR41 =2þ 3d3 =r 4 gcos 2y

ð48Þ

sy1 ¼ ð1þ lÞp=2½ð1 þ lÞpR21 =2 þd1 =r2 fð1lÞp=2 þ ½3ð1lÞpR41 =2þ 3d3 =r 4 gcos 2y

ð49Þ

try1 ¼ fð1lÞp=2 þ ½ð1lÞpR21 þ 2c1 =r2 ½3ð1lÞpR41 =2 þ3d3 =r4 gsin 2y ð50Þ

3. Solutions of stresses in the lining and the surrounding rock mass With given values of R0, R1, p, l, p0, G1, m1, G2, m2 and Z, the undetermined coefficients c1, d1, d3, e1, f1, f3, g1, g3 and h1 can be calculated by simultaneously solving the linear equation set composed by Eqs. (15)–(17), (19), (20), (23), (25)–(27). The undetermined functions j1(z), c1(z), j2(z) and c2(z), can be written as

j1 ðzÞ ¼ c1 z1 , c1 ðzÞ ¼ d1 z1 þd3 z3

ð37Þ

j2 ðzÞ ¼ e1 z1 þ f1 z þ f3 z3 , c2 ðzÞ ¼ g1 z1 þ g3 z3 þ h1 z

ð38Þ

Once j2(z) and c2(z) are known, the stresses in the lining can be obtained by Eqs. (39) and (40):

sr þ sy ¼ 4Re½j2u ðzÞ

ð39Þ 00

sy sr þ 2itry ¼ 2e2iy ½z j 2 ðzÞ þ c2u ðzÞ

ð40Þ

In order to differentiate with the stresses in the surrounding rock mass sr1, sy1 and try1, the stresses in the lining are denoted by sr2, sy2 and try2, i.e.,

If the tunnel is unsupported, c1 ¼d1 ¼d3 ¼0 in Eqs. (48)–(50). In this case, the stresses in the surrounding rock mass can be calculated by Kirsch’s solution.

4. Analysis and discussions 4.1. Distribution law of normal contact stress pn and tangential contact stress tn on the interface No hydrostatic pressure on the inner boundary of the lining was considered in Ref. [14]. In order to compare with the results in Ref. [14], the numerical sample in the reference is taken for a case study in this paper. The parameters are R1/R0 ¼1.1, m1 ¼ m2 ¼0.25, G2/G1 ¼10 and l ¼1/3. The ratios between the contact stresses determined by Eq. (44) and (1–Z)p are: p n ¼ pn =½ð1ZÞp ¼ q 1 þ q 2 cos 2y,

t n ¼ tn =½ð1ZÞp ¼ q 3 sin 2y

ð51Þ

For arbitrary p and Z, when p 40 and 0 r Z o1, it is always true that q 1 ¼ 0:3779,

q 2 ¼ 0:1798,

q 3 ¼ 0:4293

ð52Þ

sr2 ¼ 2f1 þ g1 r2 þð4e1 r2 þ 3g3 r4 h1 Þcos 2y

ð41Þ

The ratios between the contact stresses solved by Cauchy’s integral formulas in Ref. [14] and anp are given as follows:

sy2 ¼ 2f1 g1 r2 þ ð12f3 r 2 3g3 r4 þ h1 Þcos 2y

ð42Þ

q 1 ¼ 0:378,

try2 ¼ ð6f3 r 2 2e1 r2 þ 3g3 r4 þ h1 Þsin 2y

ð43Þ

By substituting r ¼R1 into Eqs. (41) and (42), the normal contact stress pn and the tangential contact stress t on the interface between the surrounding rock mass and the lining can be obtained pn ¼ q1 þq2 cos 2y,

tn ¼ q3 sin 2y

ð44Þ

q 2 ¼ 0:179,

q 3 ¼ 0:429

ð53Þ

By comparison of Eqs. (52) and (53), if three significant digits are taken, the values of q 1 are the same, q 3 has the same value but opposite sign, and q 2 has approximately the same value but opposite sign. The negative sign is caused by the different directions selected for the coordinate system. It can be seen that the unknown coefficient an introduced in [14] is actually the same as 1–Z in the present paper.

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Ai-zhong Lu et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 514–519

The distribution laws of the normal contact stress p n and the tangential contact stress t n can be obtained from Eq. (51), i.e., p n ¼ 0:37990:1798 cos 2y,

t n ¼ 0:4293 sin 2y

ð54Þ

It can be seen that the radial compressive stress and the tangential stress in the lining are functions of 2y. Owing to the symmetry, only the first quadrant is taken for analysis. The maximum radial compressive stress p nmax is 0.5577 along the x direction and the minimum value p nmin is 0.1981 along the y direction. The directions are opposite to the directions of the major and minor principal in situ stresses (the major principal in situ stress is along the y direction), respectively. From the calculation results, when G2/G1 Z85, i.e., the stiffness of the lining is much higher than that of the surrounding rock mass, the maximum radial compressive stress in the lining is no longer along the x direction, instead, it is along the y direction which is the same as the major principal in situ stress. The tangential stresses in the lining on the x and y axes are zero. When y ¼451, the tangential stress reaches the maximum value t nmax ¼0.4293, acting in the clockwise direction. From the calculation results, when 0o l o1, the distribution laws of the relative normal contact stress p n and the relative tangential contact stress t n are similar. With an increase of l, the difference between p nmax and p nmin decreases, and so does t nmax . When l ¼1, the radial compressive stress and the tangential stress on any point in the lining are the same, which are 0.5668 and 0, respectively. Similar analysis can be performed for the case of l 41. The result shows that the distribution law of the radial compressive stress is opposite to the aforementioned case and the tangential stress acts in the anticlockwise direction.

respectively. The first row is symax =p0 corresponding to y ¼901 and the second row is symin =p0 corresponding to y ¼01. It can be seen from Table 1 that: for an arbitrary in situ stress p, the smaller the displacement release coefficient Z, the larger the difference between the maximum and minimum tangential stress on the inner boundary of the lining. With the increase of Z, the difference becomes smaller. When Z ¼1, the tangential stress for any point on the inner boundary is the same and it is tensile stress. With an increase of in situ stress p, the difference between the maximum and minimum tangential stress on the inner boundary of the lining becomes larger. When Z is relatively small, the tangential stress gradually becomes compressive stress from tensile stress. 4.3. Distribution of stresses in the lining and surrounding rock In this section, the parameters are taken as: R0 ¼2.5 m, R1 ¼ 3.0 m,

m1 ¼ m2 ¼0.25, p0 ¼2 MPa, p¼4.0 MPa, l ¼1/3 and Z ¼0.4. The radial variation of normalized stresses versus normalized radial distance r/R0 in the lining and surrounding rock at y ¼01, y ¼901 with E2/E1 ¼10 and E2/E1 ¼1/10, are discussed. It can be seen from Eqs. (43) and (50) that try1 ¼ try2 ¼ 0 at y ¼01 and 901 within the lining and within the rock. The radial variation of tangential stress sy and radial stress sr at y ¼01 and 901 with E2/E1 ¼10 and E2/E1 ¼1/10, is shown in Figs. 3–6, respectively.

r/R0 1

1.2

1.5

2

2.5

3

-2 stress / p0

3.5

4

4.5

5

4

4.5

5

r1

r2

-1

4.2. Analysis of tangential stress on the inner boundary of the lining considering support delay The stress distribution in the lining is most concerned in the tunnel support design. The tangential stress reaches its maximum on the inner boundary of the lining. Owing to the symmetry, the first quadrant again is taken as an example. Eq. (42) shows that the tangential stress reaches the peak value when y is 01or 901. When 0o l o1, if tension is defined as positive, the tangential stress has its maximum value symax when y ¼901, and its minimum value symin when y ¼01. In this section, the following parameters are taken: R0 ¼2.5 m, R1 ¼3.0 m, m1 ¼0.25, m2 ¼0.20, E2/E1 ¼10.0 and l ¼1/3. The hydrostatic pressure p0 on the inner boundary of the lining is also considered and p0 ¼2 MPa. Table 1 presents symax =p0 and symin =p0 for p/p0 ¼0.5, 1.0, 2.0, 5.0, 10 and Z ¼0.0, 0.2, 0.4, 0.6, 0.8, 1.0,

1.1

0

1

-3 -4

2

-5 -6 -7 Fig. 3. Distribution of stresses (E2/E1 ¼10, y ¼01).

5 Table 1 Values of symax/p0 and symin/p0 for various p/p0 and Z. p/p0

4 3

Z 0.2

0.4

0.6

0.8

1.0

0.1

3.94 3.28

3.94 3.41

3.93 3.53

3.93 3.66

3.93 3.79

3.92

0.5

4.04 0.70

4.01 1.35

3.99 1.99

3.97 2.63

3.94 3.28

3.92

1.0

4.15  2.52

4.11  1.23

4.06 0.06

4.01 1.35

3.97 2.63

3.92

-1

2.0

4.38  8.95

4.29  6.38

4.20  3.80

4.11  1.23

4.01 1.35

3.92

-2 -3

5.0

5.08  28.26

4.85  21.82

4.61  15.38

4.38  8.95

4.15  2.51

3.92

10.0

6.23  60.43

5.77  47.56

5.31  34.69

4.85  21.82

4.38  8.95

3.92

stress / p0

0.0

2 2

1 0

1 r2 r1 1

1.1

1.2

1.5

2

2.5 r/R0

3

3.5

Fig. 4. Distribution of stresses (E2/E1 ¼10, y ¼901).

Ai-zhong Lu et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 514–519

r/R0 0

1

1.2

1.5

2

2.5

3

3.5

4

4.5

5

2

-0.5

r1

-1

r2

-1.5 stress / p0

1.1

-2 -2.5

1

-3 -3.5 -4 -4.5

519

The maximum and minimum radial compressive stresses on the interface are in the horizontal and vertical directions, respectively, which are opposite to the directions of the major and minor principal in situ stresses. With increasing amount of displacement release, the more likely the tensile tangential stress occurs on the inner boundary of the lining. This is due to the compressive stress induced by the rock deformation and the tensile stress induced by the hydrostatic pressure acting on the lining. The amount of displacement release has great influence on the tangential stress on the inner boundary of the lining. The distribution of the tangential stress can be more appropriate if the lining can be applied at a proper moment. When the in situ stress is relatively small, earlier installation of the lining is preferred. When the in situ stress is relatively large, neither too late nor too early installation of the lining is appropriate, as early installation would lead to high compressive stress in the lining and late installation would results in large tensile stress in the lining.

Fig. 5. Distribution of stresses (E2/E1 ¼ 1/10, y ¼01).

Acknowledgement 1

The study is supported by the Natural Science Foundation of China (Grant no. 50874047).

0.5

References

stress / p0

0  2

-0.5 -1

 1

r 2

-1.5 -2

r 1 1

1.1

1.2

1.5

2

2.5

3

3.5

4

4.5

5

Fig. 6. Distribution of stresses (E2/E1 ¼ 1/10, y ¼901).

It can be seen from Figs. 3–6 that, at the lining–rock interface, the radial stress sr is always continuous, and the tangential stress sy is discontinuous. With increasing radial distance r, the radial stress sr1 and the tangential stress sy1 in the rock approach the initial stress. It can be seen from Figs. 3 and 4 that, when E2/E1 ¼10, close to the inner boundary of the lining, the tangential stress is compression at y ¼01, and the tangential stress is tension at y ¼901. Their absolute value is more than the absolute value of tangential stress in the surrounding rock. It can be seen from Figs. 5 and 6 that, when E2/E1 ¼1/10, close to the inner boundary of the surrounding rock (at the lining–rock interface), the tangential stress is also compression at y ¼01, and the tangential stress is also tension at y ¼901. Their absolute value is more than the absolute value of tangential stress in the lining.

5. Conclusions When the major principal in situ stress is in the vertical direction and no hydrostatic pressure acts in the tunnel, the distribution of the contact stresses on the interface between the surrounding rock mass and the lining is the same as given by [14].

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