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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust

Imperfect bonding effect on dynamic response of a non-circular lined tunnel subjected to shear waves Xue-Qian Fang a,⇑, He-Xin Jin a, Jin-Xi Liu a, Ming-Juan Huang b a b

Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China School of Foundation, Hebei Institute of Architecture and Civil Engineering, Zhangjiakou 075000, China

a r t i c l e

i n f o

Article history: Received 24 May 2015 Received in revised form 11 January 2016 Accepted 23 March 2016

Keywords: Non-circular lined tunnel Imperfect interface Dynamic response Conformal transformation method

a b s t r a c t Combining the wave function expansion method and conformal transformation method, the dynamic stress around a non-circular tunnel with imperfect interface subjected to anti-plane shear waves is derived. The non-circular tunnel is mapped into an annular region, and the analytic solutions of stress and displacement solutions are expanded in terms of wave functions. By introducing the spring-type interface model, the coefficients are determined by satisfying the imperfect bonding conditions around the concrete lining. The distribution of tangential stresses on the imperfect interface is graphically illustrated, and the interacting effect of imperfect interface and incident wavelength is discussed in detail. The imperfect interface is revealed as a key factor dominating the seismic responses of a tunnel. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Underground tunnels are widely utilized around the world for various purposes, including subways, underground hydropower, material storage and military engineering. The tunnel liners are designed to withstand the static overburden and dynamic loading. The strength prediction is critical for the appropriate measurements, optimum design and safe operation of tunnels. In recent decades, the number of large underground tunnels has been growing significantly. Proper estimation on the stress and displacement variation under many kinds of loading is important for the optimal design of such tunnels. To ensure the safety of tunnel liners, lots of methods including theory and experimental work have been proposed to calculate the stresses and displacements in the surrounding rock and the lining. The shapes include circular tunnels (Hefny, 1999), rectangular tunnels (Lei et al., 2001), semi-circular tunnels (Exadaktylos and Stavropoulou, 2002), inverted U-shaped and notched circular tunnels (Exadaktylos et al., 2003). Although numerical methods such as finite element method (Ren et al., 2005; Wang et al., 2014) and finite difference method (Sobótka et al., 2013) have been applied to solve the response of tunnel structures, analytical techniques remain the convenient and efficient ways of providing the direct qualitative insights into ⇑ Corresponding author. E-mail address: [email protected] (X.-Q. Fang). http://dx.doi.org/10.1016/j.tust.2016.03.008 0886-7798/Ó 2016 Elsevier Ltd. All rights reserved.

the physical mechanism in the tunnel lining. For circular tunnels, the analytical solutions of displacements and stresses have been extensively studied (Fahimifar et al., 2010; Wang et al., 2012). In engineering application, the non-circular tunnel is very common. For inverted U-shaped and notched circular tunnels, Exadaktylos and Stavropoulou (2002) applied the complex potential formulation together with the conformal mapping representation to study the stresses and displacements around the tunnels with rounded corners. Kargara et al. (2014) presented a semi-analytical elastic plane strain solution for the stress field around a lined noncircular tunnel subjected to uniform ground load. The imperfect interface around the tunnels plays an important role in controlling the dynamic response of tunnels. In recent years, the analytical, experimental and numerical methods have been used to analyze the imperfect interface effect on the response of tunnels. Combining the wave function expand method and Biot’s dynamic theory of poroelasticity, Hasheminejad and Komeili (2009) studied the dynamic response of an arbitrarily thick elastic homogeneous hollow cylinder, which is imperfectly bonded to the surrounding fluid-saturated permeable formation. Based on the complex variable method, the analytical solution of stress fields of a lined non-circular tunnel with full-slip interface was presented, and the contact stresses along the rock-lining interface were analyzed (Lu et al., 2015). By using a mechanically adjustable tunnel model, Leung and Meguid (2011) designed an experimental setup to simulate the initial lining pressure that results from shield tunneling, and a local separation between the lining and the

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The steady solution is investigated, i.e., w ¼ Weixt . Then, the governing equation can be rewritten as

surrounding soil was introduced. However, no works have dealt with the imperfect interface around the non-circular tunnels subjected to dynamic loadings. The main objective of this paper is to present the analytical solutions of displacements and stresses resulting from the imperfect interfaces around the non-circular tunnels under anti-plane shear waves. Combining the wave function expansion method and conformal transformation method, the analytical solutions of anti-plane and stresses in the lining and surrounding rock are obtained. The imperfect interface around the tunnel is modeled by a spring-type medium. Through numerical examples, the effect of imperfect interface on the dynamic stress under different wave frequencies is examined.

2

@2W k þ W ¼ 0: @[email protected]z 4

where k ¼ x=cSH is the wave number in the medium. To obtain the analytical solution of this non-circular tunnel, the conformal mapping method of a complex function (shown in Fig. 2) is employed to solve the forward problem

! N X m ; z ¼ gðfÞ ¼ R f þ cm f

eih ; f ¼ n þ ig ¼ q

ð6Þ

m¼0

where n and g denote the rectangular coordinate axes in the f plane, q and h are the polar coordinate axes in the f plane. The mapping

2. Problem formulation and basis solutions

function transforms the boundaries L1 and L2 in the z-plane into two concentric circles S1 and S2 with R0 and unit radii in the f plane. Since the tunnel is symmetrical about the y-axis, the coefficients cm must be real numbers. All coefficients (R and cm ) in the mapping function can be determined by optimization methods when the shape of the tunnel cross-section and support thickness are known (Lu, 1996). The value of N is related to the shape of the tunnel, and the truncation at 4 is enough for computation. From Eq. (3), the shear stress in the f plane can be expressed, in the cylindrical coordinate system, as

In this paper, a lined non-circular tunnel with imperfect interface is considered, as depicted in Fig. 1. The tunnel is embedded in the rock mass, and it is assumed that the surrounding rock mass and the lining are both linear isotropic and elastic. These two isotropic homogenous regions (S1 and S2) are bounded by contours L1 and L2. An anti-plane shear wave with frequency of x propagates in the rock mass. Due to the geometrical character and the character of applied loading, this problem can be simplified into a plane strain problem. The shear modulus and density of rock mass are denoted by lR and qR , and those of the lining are lL and qL . For this anti-plane problem, only the out-of-plane displacement field is considered, i.e.,

ux ¼ uy ¼ 0;

ð5Þ

uz ¼ wðx; y; tÞ;

srz ¼

@W @W þf ; f @z @z 2

l

shz ¼

il @W @W f : f @z @z 2

ð7Þ

where f denotes the conjugate of variable f. To obtain the analytical expression in the rock mass and lining, Eq. (5) is transformed into the equation about variables f and f,

ð1Þ

where ux , uy and uz denote the displacements in the x, y and z directions. The governing equation in the isotropic homogenous medium is described as

@2W k þ W ¼ 0; g 0 ðfÞg 0 ðfÞ @[email protected]f 4

@2w @2w 1 @2w þ 2 ¼ 2 ; 2 @x @y csh @t2

where X 0 the prime denotes the derivative of X, and X is the conjugate of X.

ð2Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ where cSH with cSH ¼ l=q is the wave speed of anti-plane shear waves, and t is the time. The constitutive relations of anti-plane shear displacement are expressed as

sxz ¼ l

@w ; @x

syz ¼ l

@w ; @y

2

1

ð8Þ

3. Wave fields and stresses in the rock mass and concrete lining 3.1. The incident wave and corresponding stress The anti-plane shear waves with incident angle of a propagate in the rock mass. It is convenient to express the displacement in the cylindrical coordinate system, i.e.,

ð3Þ

To express the wave field around the non-circular tunnel, the complex variable z ¼ x þ iy and its complex conjugate z ¼ x iy should be introduced. Then, the following relations can be obtained

W ðinÞ ¼ W 0

@ @ @ ¼ þ ; @x @z @z

where W 0 is the amplitude of incident waves, kR is the wave number in the rock mass, and J n ðÞ is the nth Bessel function of the first kind.

@ @ @ ¼i ; @y @z @z

Tunnel

SH waves

S2

L1 o

ð9Þ

η Imperfect interface

Rock mass

L2

Imperfect interface

n

i J n ðkR jgðfÞjÞeinðhaÞ ;

n¼1

ð4Þ

y

1 X

x

ρ

1 o S1

θ R0

ξ

α (a)

(b)

Fig. 1. A lined non-circular tunnel with imperfect interface under anti-plane shear waves. (a) Lined non-circular tunnel in the z plane; (b) Ring-shaped region in the f plane.

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y SH waves 120°

40°

o 220°

4.4m

x

7.6m

0.3m

jgðfÞj

7.3m

sðrlÞ rz ¼

320°

sðrlÞ hz ¼

sðinÞ hz ¼

i fnJn ½kR jgðfÞj kR jgðnÞjJ nþ1 ½kR jgðfÞjgeinðhaÞ ; ð10Þ n

n¼1 1 X n RW0

l i nJn ½kR jgðfÞjeinðhaÞ : jxðfÞj n¼1

ð11Þ

When the incident waves run into the lined tunnel, the scattered waves come into being. The displacement field of the scattered waves is given by 1 X

ð12Þ

n o ð1Þ ð1Þ An Hn1 ½kR jgðnÞjHnþ1 ½kR jgðnÞj einh ;

1 lR kR X

2

n¼1 1 X k R R

ð1Þ

ð1Þ

An fHn1 ½kR jgðfÞj þ Hnþ1 ½kR jgðfÞjgeinh :

ð13Þ ð14Þ

n¼1

3.3. The refracted field and the stress inside the concrete lining The refracted waves, being confined inside the concrete, are standing waves, and represented by

W ðrrÞ ¼

1 X

inh Bn Hð2Þ n ðkL jgðfÞjÞe ;

ð15Þ

n¼1

where Bn are the mode coefficients of refracted waves around the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ tunnel, kL ¼ x=cL with cL ¼ lL =qL is the wave number in the conHð2Þ n ðÞ

is the nth Hankel function of the second kind, crete lining, and denotes the inwards propagating waves. The stress produced by the refracted waves can be written as

sðrrÞ rz ¼ sðrrÞ hz ¼

1 lL kL X

2

n o ð2Þ ð2Þ Bn Hn1 ½kL jgðfÞjHnþ1 ½kL jgðfÞj einh ;

n¼1 1 X L kL

il 2

ð1Þ

ð19Þ

1 i lL k L X ð1Þ ð1Þ C n fHn1 ½kL jgðfÞj þ Hnþ1 ½kL jgðfÞjgeinh : 2 n¼1

ð20Þ

ðtÞ

W R ¼ W ðinÞ þ W ðscÞ :

ð21Þ

Similarly, the total stress field in the rock mass is expressed as

ð22Þ ð23Þ

The total wave field in the concrete lining is produced by the superposition of the refracted waves and the reflected waves resulting from the inner boundary of lining, i.e.,

W L ¼ W ðrrÞ þ W ðrlÞ :

ð24Þ

The total stress field in the concrete lining is expressed as

where An are the mode coefficients of scattered waves around the

il 2

ð1Þ

C n fHn1 ½kL jgðfÞj Hnþ1 ½kL jgðfÞjgeinh ;

n¼1

ðtÞ

An Hnð1Þ ðkR jgðnÞjÞeinh ;

tunnel, Hð1Þ n ðÞ is the nth Hankel function of the first kind, and denotes the outgoing propagating waves. The stress produced by the scattered waves can be written as

sðscÞ hz1 ¼

2

The total wave field in the rock mass is produced by the superposition of the incident and scattered waves resulting from the lining, i.e.,

n¼1

sðscÞ rz ¼

1 lL kL X

ðtÞ ðinÞ srzR ¼ srz þ sðscÞ rz : ðtÞ ðinÞ shzR ¼ shz þ sðscÞ hz :

3.2. The scattered field and stress in the rock mass

W ðscÞ ¼

ð18Þ

where C n are the mode coefficients of reflected waves in the concrete lining. The dynamic stress produced by the reflected waves can be written as

The stresses resulting from the incident waves can be written as 1 lR W 0 X

inh C n Hð1Þ n ðkL jgðfÞjÞe ;

n¼1

Fig. 2. The geometry of the tunnel and lining in numerical examples.

sðinÞ rz ¼

1 X

ðrlÞ

W2 ¼

ð2Þ

ð2Þ

Bn fHn1 ½kL jgðfÞj þ Hnþ1 ½kL jgðfÞjgeinh :

ð16Þ ð17Þ

n¼1

3.4. The reflected field and stress in the concrete lining There are reflected waves that propagate outwards from the inner boundary of concrete. The displacement field of the reflected waves is given by

ðtÞ ðrlÞ srzL ¼ sðrrÞ rz þ srz : ðtÞ hzL

s

ðrrÞ hz

¼s

þs

ð25Þ

ðrlÞ hz :

ð26Þ

4. Boundary conditions To analyze the imperfect interface effect, a spring-type interface model is introduced. In this model, the tractions at the outer boundary of the concrete are continuous, but the displacements are discontinuous across the interface (Shen et al., 2001; Valier-Brasier et al., 2012). The normal tractions are proportional to the corresponding displacement discontinuities through stiffness parameters. The limitation of this interface model is that the detachment of lining from surrounding rock is not allowed. Using this model, the boundary conditions at the interfaces of the concrete lining (L2 in Fig. 1) can be described as follows, ðtÞ ðtÞ srzR ¼ srzL ; ðtÞ

ðtÞ

WR WL ¼

ð27Þ

s

ðtÞ rzR

d

;

ð28Þ

where d is the stiffness parameter of the imperfect interface. It represents the elastic property of interface, and can be measured by ultrasonic method. The stiffness parameter in the interface model represents the elastic property of interface. For d ! 1, with the tangential stress being finite quantities, the anti-plane displacement is continuous at the interface. The interface model therefore ðtÞ

approaches a perfectly bonded interface. For d ! 0, srzR should be zero, which means that no waves are transmitted from the rock mass to the tunnel liner. At the inner boundary of concrete lining (L1 in Fig. 1), it is assumed that the tractions are free. It can be expressed as ðtÞ srzL ¼ 0;

ð29Þ

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Substituting Eqs. (21)–(26) into Eqs. (27)–(29) and making use of the orthogonality of eish , a set of linear algebra equation can be obtained as follows,

0 6 4 X 21 X 31

X 12 X 22 X 32

32

3

2

3

As 0 76 7 6 7 X 23 54 Bs 5 ¼ 4 Y 21 5: X 33 Cs Y 31 X 13

R

R0 (m)

c1

c2

c3

c4

4.33

0.93

0.0246

0.0834

0.0521

0.0418

ð30Þ 1.8

The elements in this matrix equation can be found in the Appendix A. The coefficients As , Bs , C s can be determined by solving Eq. (30). During computation, it is found that the truncation after s = 8 gives practically adequate results at any desired frequency. It is noted that the truncation of s is only related to the incident wave frequency, and a smaller truncated number can be adopted when the wave frequency is smaller.

1.2

0.8

0.2 0

0

40

80

120

160

θ(˚)

200

240

280

360

Fig. 3. Comparing with the existing results (A circular tunnel with perfect interface). 1. k ¼ 0:1, obtained from Pao and Mao (1973); 2. k ¼ 0:1, obtained from this paper; 3. k ¼ 0:5, obtained from Pao and Mao (1973); 4. k ¼ 0:5, obtained from this paper.

1 ilR kR X ð1Þ ð1Þ An fHn1 ½kR jgðfÞj þ Hnþ1 ½kR jgðfÞjgeinh : 2 n¼1

ð32Þ

In the following numerical examples, the material properties of rock mass and concrete lining illustrated in Table 1 are used. The geometries of the tunnel and the lining are shown in Fig. 2. The main objective of this paper is to investigate the imperfect interface effect, and the incident angle shows no effect on the imperfect interface. So, the incident angle is assumed to be zero, i.e., a ¼ 0. Coefficients of conformal mapping function in Eq. (6) are given in Table 2. Some dimensionless parameters are adopted, i.e., k ¼ kR jxðfÞj ¼ 0:1 1:5 and d ¼ djxðfÞj=lR ¼ 0:1 50:0 (Yi et al., and R0 are m. In these figures, h ¼ 0 2014). The units of W 0 , q corresponds to the x axis, and h ¼ 90 corresponds the y axis above the midpoint of the vault, as shown in Fig. 2. To validate the present elastodynamics model, comparison with the existing solutions is given. Fig. 3 illustrates the DSCF around a circular tunnel with perfect interface. The numerical results are consistent with those obtained from Pao and Mao (1973). When k ¼ 0:1, the maximum dynamic stress occurs at the positions of h ¼ 90 and 270 . If k ¼ 0:5, the maximum dynamic stress is at the h ¼ 40 and 320 .

Fig. 4 shows the dynamic stress distribution around the tunnel in the region of low frequency. Similar with the circular tunnel, the maximum dynamic stress occurs at the positions near h ¼ 40 and 320 . The interface effect on the dynamic stress at the positions near h ¼ 75 ; 200 is the greatest, and the dynamic stress decreases with increasing stiffness parameter of interface. h ¼ 75 corresponds to the positions near the top the vault. At the positions near h ¼ 0 and h ¼ 180 , the interface parameter expresses no effect on the dynamic stress. So, a smaller stiffness parameter is proposed for the tunnels serving in the low frequency loadings. Fig. 5 shows the dynamic stress distribution around the tunnel in the region of intermediate frequency. The maximum dynamic stress occurs at the positions near h ¼ 60 and 310 . By comparing with the results in the region of low frequency, the interface effect increases greatly, especially at the positions near h ¼ 60 , 100 and 250 . It is interesting to note that the dynamic stress at the bottom increases with increasing stiffness parameter of interface, however, the dynamic stress at the positions above the vault decreases with

1.6 1.4 1.2

DSCF

n

i nJ n ½kR jgðfÞjeinðhaÞ

n¼1

1

0.8 0.6 0.4

1 2 3

0.2

Table 1 Material properties of the model. Elastic properties of rock

320

ð31Þ

where s0 is the maximum magnitude of stress in the incident direction, and s0 ¼ W 0 lR kR . The circumferential stress in terms of incident wave potential is expressed as

þ

4

1

0.4

DSCF ¼ shz ¼ jshz =s0 j:

jgðfÞj

3

0.6

To analyze the imperfect interface effect on the dynamic response of the non-circular tunnel under different loading frequencies, the dimensionless circumferential stress is introduced. According to the definition of the dynamic stress concentration factor (DSCF), the DSCF is the ratio of the circumferential stress shz around the inclusion and the maximum dynamic stress (Pao and Mao, 1973). Thus, the DSCF around the non-circular tunnel is expressed as

1 lR W 0 X

2

1.4

5. Numerical examples and analysis

sðtÞ hzR ¼

1

1.6

DSCF

2

Table 2 Coefficients in the conformal mapping function.

Elastic properties of concrete lining

lR (Gpa)

qR (kg/m3)

lL (Gpa)

qL (kg/m3)

40

2.7 103

25

2.5 103

0 0

40

80

120

160

θ(˚)

200

240

280

320

360

Fig. 4. Dynamic stress around the non-circular tunnel along L2 under different interface properties in the region of low frequency (k ¼ 0:5). 1. d ¼ 0:5; 2. d ¼ 5:0; 3. d ¼ 20:0.

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1.6

anti-plane waves, and the analytical solutions are derived. The new elastodynamic model improved previous solutions, and the numerical solutions are illustrated to analyze the interface effect on the dynamic stress around the tunnel. To improve the strength of tunnels, a smaller stiffness parameter of interfaces at the floor is proposed. Under different loading frequencies, the interface effect at the vault will show great variation. The solving method can provide significant reference for the design and construction of underground tunnels.

1.4

DSCF

1.2 1 0.8 0.6 0.4

1 2 3

0.2 0 0

40

80

120

160

θ(˚)

200

240

280

Acknowledgements

320

360

Fig. 5. Dynamic stress around the non-circular tunnel along L2 under different interface properties in the region of intermediate frequency. (k ¼ 1:0) 1. d ¼ 0:5; 2. d ¼ 5:0; 3. d ¼ 20:0.

The paper is supported by National Natural Science Foundation of China (No. 11472181). Appendix A

X 12 ¼

1.5

X 13 X 21

1

DSCF

X 22

lL kL

ð2Þ

ð2Þ

fHs1 ½kL jgðfÞj Hsþ1 ½kL jgðfÞjg; 2 l kL ð1Þ ð1Þ ¼ L fHs1 ½kL jgðfÞj Hsþ1 ½kL jgðfÞjg; 2 lR kR ð1Þ ð1Þ fHs1 ½kR jgðfÞj Hsþ1 ½kR jgðfÞjg; ¼ Hð1Þ s ½kR jgðfÞj 2d ¼ Hð2Þ s ½kL jgðfÞj;

X 23 ¼ Hð1Þ s ½kL jgðfÞj; X 31 ¼

0.5

40

80

120

160

θ(˚)

200

240

280

2

X 32 ¼

1 2 3

0 0

lR kR

X 33 320

360

Fig. 6. Dynamic stress around the non-circular tunnel along L2 under different interface properties in the region of high frequency (k ¼ 1:5) 1. d ¼ 0:5; 2. d ¼ 5:0; 3. d ¼ 20:0.

increasing stiffness parameters of interface. The dynamic stress at the floor increases with increasing stiffness parameter of interface. To reduce the dynamic stress, a small stiffness parameter of interface at the floor and illuminate sides is preferred, and a large stiffness parameter near the vault is proposed. Fig. 6 shows the dynamic stress distribution around the tunnel in the region of high frequency. It can be seen that more peaks occur around the tunnel due to the high frequency loading. The dynamic stress decreases significantly if a small stiffness parameter of interface is selected, especially at the positions near the vault and the floor. The interface effect increases significantly if the loading frequency is high. The variation of dynamic stress with stiffness parameter is greater at the positions near the vault and at the floor. This phenomenon results from the strong scattering of elastic waves at these positions. If the tunnel is serving under loadings with high frequency, a smaller stiffness parameter is proposed. By comparing the results in Figs. 4–6, it can be concluded that the interface effect is significant if the value of d is less than 1.0. The interface effect also increases with increasing wave frequency of loadings. 6. Conclusions An imperfect interface model is introduced to analyze the dynamic stress distribution around a non-circular tunnel under

Y 21

ð1Þ

ð1Þ

fHs1 ½kR jgðfÞj Hsþ1 ½kR jgðfÞjg;

lL kL

ð2Þ

ð2Þ

fHs1 ½kL jgðfÞj Hsþ1 ½kL jgðfÞjg; 2 l kL ð1Þ ð1Þ ¼ L fHs1 ½kL jgðfÞj Hsþ1 ½kL jgðfÞjg; 2 l W0 s ¼ R i fsJs ½kR jgðfÞj kR jgðfÞjJ sþ1 ½kR jgðfÞjgeisa jgðfÞjd s

Y 31

W 0 i J s ½kR jgðfÞjeisa ; l W0 s ¼ R i fsJ s ½kR jgðfÞj kR jgðfÞjJ sþ1 ½kR jgðfÞjgeisa : jgðfÞj

References Exadaktylos, G.E., Liolios, P.A., Stavropoulou, M.C., 2003. A semi-analytical elastic stress–displacement solution for notched circular openings in rocks. Int. J. Solids Struct. 40 (5), 1165–1187. Exadaktylos, G.E., Stavropoulou, M.C., 2002. A closed-form elastic solution for stresses and displacements around tunnels. Int. J. Rock Mech. Min. Sci. 39 (7), 905–916. Fahimifar, A., Farshad, M.T., Ahmadreza, H., Arash, V., 2010. Analytical solution for the excavation of circular tunnels in a visco-elastic Burger’s material under hydrostatic stress field. Tunn. Undergr. Space Technol. 25 (4), 297–304. Hasheminejad, S.M., Komeili, M., 2009. Effect of imperfect bonding on axisymmetric elastodynamic response of a lined circular tunnel in poroelastic soil due to a moving ring load. Int. J. Solids Struct. 46, 398–411. Hefny, A.M., 1999. Analytical solutions for stresses and displacements around tunnels driven in cross-anisotropic rocks. Int. J. Num. Anal. Met. 23 (2), 161– 177. Kargara, A.R., Rahmannejada, R., Hajabasi, M.A., 2014. A semi-analytical elastic solution for stress field of lined non-circular tunnels at great depth using complex variable method. Int. J. Solids Struct. 51 (6), 1475–1482. Lei, G.H., Ng, C.W.W., Rigby, D.B., 2001. Stress and displacement around an elastic artificial rectangular hole. J. Eng. Mech. ASCE 127 (9), 880–890. Leung, C., Meguid, M.A., 2011. An experimental study of the effect of local contact loss on the earth pressure distribution on existing tunnel linings. Tunn. Undergr. Space Technol. 26, 139–145. Lu, A.Z., 1996. Shape optimization based on the criterion of making the absolute maximum tangential stress of the hole be the minimum. Chin. J. Solid. Mech. 17 (1), 73–76. Lu, A.Z., Zhang, N., Qin, Y., 2015. Analytical solutions for the stress of a lined noncircular tunnel under full-slip contact conditions. Int. J. Rock Mech. Min. Sci. 79, 183–192.

X.-Q. Fang et al. / Tunnelling and Underground Space Technology 56 (2016) 226–231 Pao, Y.H., Mow, C.C., 1973. Diffraction of Elastic Waves and Dynamic Stress Concentrations. Crane, Russak, New York. Ren, G., Smith, J.V., Tang, J.W., Xie, Y.M., 2005. Underground excavation shape optimization using an evolutionary procedure. Comput. Geotech. 32 (2), 122– 132. Shen, H., Schiavone, P., Ru, C., Mioduchowski, A., 2001. Stress analysis of an elliptic inclusion with imperfect interface in plane elasticity. J. Elast. Phys. Sci. Solids 62 (1), 25–46. Sobótka, M., Łydzba, D., Rózan´ski, A., 2013. Shape optimization of underground excavation by simulated annealing. Stud. Geotech. Mech. 35 (1), 209–218.

231

Valier-Brasier, T., Dehoux, T., Audoin, B., 2012. Scaled behavior of interface waves at an imperfect solid-solid interface. J. Appl. Phys. 112 (2), 024904. Wang, T.T., Hsu, J.T., Chen, C.H., Huang, T.H., 2014. Response of a tunnel in doublelayer rock s subjected to harmonic P- and S-waves. Int. J. Rock Mech. Min. Sci. 70, 435–443. Wang, S.L., Wu, Z.J., Guo, M.W., Ge, X.R., 2012. Theoretical solutions of a circular tunnel with the influence of axial in situ stress in elastic-brittle-plastic rock. Tunn. Undergr. Space Technol. 30, 155–168. Yi, C.P., Zhang, P., Daniel, J., Ulf, N., 2014. Dynamic response of a circular lined tunnel with an imperfect interface subjected to cylindrical P-waves. Comput. Geotech. 55, 165–171.