Simple approach for solution of the quasi-plane-strain problem in a circular tunnel in a strain-softening rock mass considering the out-of-plane stress effect

Simple approach for solution of the quasi-plane-strain problem in a circular tunnel in a strain-softening rock mass considering the out-of-plane stress effect

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Journal Pre-proofs Simple approach for solution of the quasi-plane-strain problem in a circular tunnel in a strain-softening rock mass considering the out-of-plane stress effect Zou Jin-feng, Liu Lu, Xia Ming-yao PII: DOI: Reference:

S2467-9674(19)30082-0 https://doi.org/10.1016/j.undsp.2019.09.003 UNDSP 117

To appear in:

Underground Space

Please cite this article as: Z. Jin-feng, L. Lu, X. Ming-yao, Simple approach for solution of the quasi-plane-strain problem in a circular tunnel in a strain-softening rock mass considering the out-of-plane stress effect, Underground Space (2019), doi: https://doi.org/10.1016/j.undsp.2019.09.003

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Simple approach for solution of the quasi-plane-strain problem in a

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circular tunnel in a strain-softening rock mass considering the

4

out-of-plane stress effect

5 6

Zou Jin-feng1, Liu Lu2 and Xia Ming-yao3,*

7 8 9

1School of Civil Engineering, Central South University, Changsha 410075, China

E-mail: [email protected]

10 11

2School of Civil Engineering, Central South University, Changsha 410075, China

E-mail: [email protected]

12 13

3School of Civil Engineering, Central South University, Changsha 410075, China

E-mail: [email protected]

14 15 16 17

*Corresponding

authort. Tel.: +86-138-73105859 E-mail: [email protected]

Abstract: The out-of-plane stress is sometimes the major or intermediate principal stress in a circular tunnel opening. The influences of the out-of-plane stress and axial strain are often neglected in the stability analyses of tunnel excavation, which can induce significant errors in the determination of surrounding rock deformations. In this paper, the use of a simple approach is proposed to solve the quasi-plane-strain problem of circular tunneling considering the effect of the out-of-plane stress, which is deformation-dependent and influenced by the in situ stress. As the intermediate principal stress is deformation-dependent, to obtain the numerical solution of the intermediate principal stress, the quasi-plane-strain problem is defined based on assumptions that the initial axial total strain is a nonzero constant () and that the axial plastic strain is nonzero. With the numerical solution for the plastic strain, obtained using the plastic potential functions based on the three-dimensional failure criteria, the formula for the intermediate principal stress can be derived using Hooke’s law. The proposed approach can be utilized to obtain the numerical solution for the intermediate principal stress, which is deformation-dependent, and the numerical results can be simplified as the solution presented by Pan and Brown. The proposed approach can also be used to obtain the solution for the strain softening of the surrounding rock. To verify its validity and accuracy, the results obtained using the proposed approach are compared with the solution of Pan and Brown. In addition, parametric studies are performed to address the influences of the out-of-plane stress on the stress and displacement in the circular tunnel.

1

Keywords:

Quasi-plane-strain

problem;

Plastic

strain;

Out-of-plane

stress;

Three-dimensional plastic potential function; Deformation dependence

1 Introduction It is well known that the stability of a tunnel is a major issue during tunnel excavation. Currently, a majority of the analyses are based on the plane strain assumption, wherein the intermediate principal stress and strain are not considered (Alonso et al., 2003; Brown et al., 1983; Carranza and Fairhurst, 1999;Carranza, 2003; Carranza, 2004; Lee and Pietruszczak, 2008; Park et al., 2008; Sharan, 2008; Wang et al., 2010; Wang et al., 2011; Wang et al., 2012; Wang et al., 2012; Zou and Qian, 2018; Zou et al., 2018). Although the plane strain assumption is widely accepted, superior results can be obtained by considering the influence of the out-of-plane stress and strain, particularly in the case of the strain softening of the surrounding rock. Brown et al. (1983) presented a closed-form solution for the elastic–perfectly plastic and elastic-brittle–plastic rock masses based on the simple axisymmetric assumption and generalized Hoek–Brown (H–B) failure criterion, but the plastic strain variable in the plastic zone was not considered in the strain softening analysis. Carranza-Torres et al. (1999, 2003, 2004) combined the elastic–perfectly plastic model with the nonlinear H–B failure criterion. The solutions proposed by Carranza-Torres et al. were more accurate and can be used to validate other approximation solutions because of the fewer simplifications used in the derivations of the former. However, the solutions proposed by Carranza-Torres et al. were significantly limited by the differences between the ideal elastic–plastic model and 2

stress–strain properties of the rock mass in actual applications. Sharan (2008) obtained analytical solutions for the stress and displacement based on the plane strain assumption and generalized H–B failure criterion. Park et al. (2008) proposed an improved solution, in which the elastic strain and dilation angle in the plastic region are considered as variable parameters. Park et al. (2014) presented a similar solution for a spherical or circular opening excavated in an elastic strain-softening rock mass. This solution is compatible with either a linear Mohr–Coulomb (M–C) or nonlinear H–B yield criterion. Alonso et al. (2003) proposed a new method for the generation of ground response curves for the softening of the surrounding rock considering the nonassociated flow rule and various failure criteria, such as the Tresca, M–C, and H– B failure criteria. Moreover, Lee and Pietruszczak (2008) introduced a numerical procedure for the softening of the surrounding rock. However, in the above studies, the out-of-plane stress was not considered in the solutions. Wang et al. (2010) proposed a new approach for the analysis of circular tunnel excavation in a strain-softening rock. This approach is based on the salient characteristic of the strain-softening media, whose strength parameters decrease in the post-peak stage. Wang et al. (2011) developed a procedure for the modelling of the strain-softening behavior based on the methodology for the analysis of a brittle– plastic rock mass, which follows the M–C yield criterion. Wang et al. (2012) proposed a closed-form solution for the elastic-brittle–plastic and strain-softening rock masses in a spherical cavity. Zou and Li (2015) proposed an improved numerical method for the analysis of the stability of a strain-softening surrounding rock

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considering the hydraulic-mechanical coupling and variation in elastic strain in the plastic region. In the above reports, the analytical and numerical solutions based on the plane strain assumption are discussed using different constitutive models (e.g., elastic–plastic, elastic-brittle–plastic, and strain-softening models) combined with the linear M–C or generalized H–B failure criteria. Although a few studies (Lu et al., 2010; Pan and Brown, 1996; Reed, 1988; Wang et al., 2012b; Zou et al., 2011; Zou et al., 2015) have been focused on the effect of the out-of-plane stress, the assumptions of zero axial plastic strain and simple plastic potential function were employed in the majority of the studies. For example, Reed (1988) discussed the stress and displacement in a circular tunnel in M–C media considering the axial stress. Pan and Brown (1966) reported that the axial plastic strain should be nonzero when the out-of-plane stress was considered as the intermediate stress and proposed a numerical solution for an elastic-brittle–plastic rock mass. Lu et al. (2010) and Zhou et al. (2011) assumed that the initial axial strain was constant and that increased slightly after the excavation and performed an elastic– plastic analysis using the M–C as well as the H–B failure criteria. Theoretical solutions for the stress and displacement were proposed by Wang et al. (2012) considering the out-of-plane stress as the major, intermediate, and minor principal stress. Zou et al. (2015) presented theoretical solutions for the stress, displacement, and plastic radius for an elastic-brittle–plastic rock mass based on the generalized H– B failure criterion. Zou and Su (2016) developed a solution in which the effects of the out-of-plane stress and seepage force were considered for an elastic-brittle–plastic or

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elastic–plastic rock mass in combination with the M–C and generalized H–B failure criteria. Although Pan and Brown (1996) proposed that the axial in situ stress in the plastic zone is deformation-dependent and the formula for the calculation of the intermediate principal stress was derived, only a numerical solution was presented using the finite-element method. In addition, when the out-of-plane stress was the major principal stress, the influences of the out-of-plane stress on the stress and displacement were not considered. The main objective of this study was to introduce a new approach for the mechanical analysis of a circular tunnel by considering the influence of the out-of-plane stress based on the assumption of the quasi-plane-strain problem of tunnel. The corresponding theoretical solutions for the intermediate principal stress, which is deformation-dependent, and axial strain are obtained.

2 Methodology 2.1 Definition of the problem Figure 1 shows a circular opening with a radius of r0 in an initially elastic medium. The excavation surface is subjected to a hydrostatic pressure (p0) at the cross section of the tunnel and out-of-plane stress (q) along the axis of the tunnel. An internal support pressure (pin) acts uniformly on the tunnel wall surface in the radial direction after the excavation. The stress and displacement of the surrounding rock depend only on the radius in the cylindrical polar coordinate system without considering the gravity field. r,  and z are the radial, circumferential, and axial 5

stresses, respectively, R is the plastic radius, Rs is the plastic radius for the softening region, and Rf is the radius of the interface of the inner and outer plastic regions.

Fig. 1. Model of the axisymmetric and quasi-plane-strain problem for a tunnel. 2.2 Assumptions To derive the solution for the circular tunnel opening considering the influences of the out-of-plane stress and axial strain, the following assumptions are employed: (1) The surrounding rock of the circular tunnel opening is considered to be continuous, homogeneous, isotropic, and initially elastic. Based on the quasi-plane-strain assumption, the use of a new simple theoretical approach is proposed considering the effect of the out-of-plane stress, which is deformation-dependent and influenced by the in situ stress. (2) According to the Pan and Brown’s (1996) study, the axial plastic strain should be nonzero and should be used to evaluate the axial stress. To obtain

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the

numerical

solution

for

the

intermediate

principal

stress,

the

quasi-plane-strain problem is defined based on the assumptions that the initial axial total strain is a nonzero constant (ε0) and that the axial plastic strain is nonzero. (3) For the quasi-plane-strain problem, the deformations in the plastic and elastic regions are based on the small-strain assumption. The M–C and generalized H–B failure criteria are used for the stresses in the rock in the plastic state. 2.3 Failure criterion The yielding function of the rock mass is expressed by F ( 1 ,  3 ,  p )   1   3  H ( 1 ,  p ) ,

(1)

where  1 and  3 are the major and minor principle stresses, respectively, H is the function of principle stress and strain-softening parameter, and  p is the deviatoric plastic strain, which controls the evolution of the strain-softening parameter in the softening region and is generally expressed as  p = 1p   3p , where  1p and  3p are the major and minor deviatoric strains, respectively. For the M–C failure criterion, H in Eq. (1) is H MC ( 3 ,  p )  ( N ( p )  1) 3  Y ( p ) ,

(2)

where N and Y are the strength parameters defined by the cohesion c( p ) and internal friction angle  ( p ) , respectively. N ( p )=

1  sin    p  1  sin    p 

; Y ( p ) 

2c   p  cos    p  1  sin    p 

.

(3)

If the generalized H–B failure criterion is employed, H in Eq. (1) can be expressed as 7

H

HB

  3 ( 3 ,  )   c ( )  m   p   s  p   p  c ( )   p

 

a p

p

,

(4)

where  c is the uniaxial compressive strength of the intact rock and a , m , and s are the strength parameters of the generalized H–B failure criterion. 2.4 Critical internal pressure of the surrounding rock (pci) The internal support pressure (pin) in the critical state is related to the orders of magnitude of σ z , σθ , and σ r . σ θ pin  (0, p0 ) and

is always larger than σ r

as long as

σ r is often the minor principal stress in the plastic zones [9]. As the

surrounding rock is in the elastic–plastic critical state,  r |r  r0  pin and

σz = q .

When the out-of-plane stress along the axis of the tunnel is the major principal stress, the critical internal pressure (pc1) can be calculated using Eq. (1). In the case of the M–C failure criterion, Eq. (1) can be expressed as pc1MC  (q  Yp ) / N p ,

(5-a)

where Np and Yp are the critical strength parameters at internal face. In the case of the generalized H–B failure criterion, Eq. (1) can be expressed as q  pc1HB   c (m

pc1HB

c

 s)a .

(5-b)

When the out-of-plane stress is the intermediate principal stress, the critical internal pressure (pc2) can be determined using Eq. (1). In the case of the M–C failure criterion, Eq. (1) can be expressed as pc2MC  (2 p0  Yp ) / (1  N p ) .

(6-a)

In the case of the H–B failure criterion, Eq. (1) can be expressed as 2 p0  pc2HB  pc2   cp (mp 8

pc2HB

 cp

 sp )

ap

,

(6-b)

when the surrounding rock mass is in the elastic–plastic critical state and r  r0 , pin =pc2 ,

 r  pin , and

   2 p0   r  2 p0  pc2 . If we define

q1  pc2 and

q2  2 p0  pc2 , then q is the major and intermediate principal stress (i.e., q  q2 and q2  q  q1 ). It should be noted that when the plastic region is formed, the radial stress (  R ) acting on the elastic–plastic interface is equal to pci (i.e.,  R   r ( R)  pci ). 2.5 Plastic potential function The plastic potential functions based on the three-dimensional M–C and generalized H–B failure criteria are employed in this study to obtain the solution for the strains. Based on the M–C failure criterion, the plastic potential function can be expressed as

Q    J 2   I1 , where I1   1   2   3 , J 2 

(7)

1 2 2 2  1   2    2   3    3   1   ,  1 ,  2 , and  6

 3 are the major, intermediate, and minor principal stresses, respectively, and  is the dilation parameter expressed in the Pan and Brown’s study (1996).

 max 

2sin  N 1  , 3  3  sin   3  N  2

where max is the maximum value of the dilation parameter  and N 

(8)

1  sin  is 1  sin 

the triaxial stress factor. When the generalized H–B criterion is employed, the plastic potential function can be expressed by

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n 3 3 Q     I1  J2  n J2 , 3 c 2

(9)

where n is the dilation parameter. Based on the plastic flow rule, the plastic strain increment can be expressed as d p  d

Q , 

(10)

where Q is the plastic potential, d is the plastic constant, and d p is the plastic strain increment. In the case of the M–C failure criterion, the increments in the major, intermediate, and minor plastic strains can be expressed as   2 1   2   3   d1p      d , 6 J2  

(11-a)

  2 2   1   3   d 2p      d , 6 J2  

(11-b)

   2 3   1   2     d . d 3p   6 J2  

(11-c)

In the case of the generalized H–B failure criterion, the increments in the major, intermediate, and minor plastic strains can be expressed as  3  2 1   2   3  1   1   n   2 1   2   3   d , d1p   c 3  12 J 2  

(12-a)

 3  2 2   3   1  1   1   n   2 2   3   1   d , d 2p   c 3  12 J 2  

(12-b)

 3  2 3   2   1  1   1   n   2 3   2   1   d , d 3p   c 3  12 J 2  

(12-c)

p p p where 1 ,  2 , and  3 are the major, intermediate, and minor principal strains,

respectively.

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2.6 Types of plastic zones around the tunnel The plastic zone is divided into different regions in the softening surrounding the rock when the out-of-plane stress is considered. Considering the strain-softening behavior of the rock mass, the plastic region can be divided into the softening and residual regions with plastic radii denoted as R and Rs , respectively. As discussed in the literature (Lu et al., 2010; Pan and Brown, 1996; Reed, 1988), owing to the influence of the intermediate principal stress, there are two potential types of plastic zones with various orders of stresses: (I) when the out-of-plane stress is the major principle stress, see Fig.2 (a); case (1) wherein

 z      r and case (2) comprising the inner plastic region wherein  z      r and outer plastic region wherein  z      r and (II) when the out-of-plane stress is the intermediate principle stress, see Fig.2 (b); case (1) wherein     z   r and case (2) comprising the inner plastic region wherein     z   r and outer plastic region wherein     z   r . Schematics for these two types of plastic zones are presented in Fig. 2.

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Fig. 2. Types of plastic zone. (a), (b) Evolutions of the plastic types I and II, respectively. 2.7 Deterioration of the strength and deformation parameters The strength and deformation parameters of a strain-softening rock mass are evaluated using the plastic deformation, controlled by the deviatoric strain.

 p   1p   3p , p

(13)

p

where γ1 and γ3 are the major and minor deviatoric strains, respectively. The physical parameters of the surrounding rock are expressed using the bilinear function of the plastic shear strain.  γp p p       p  p r  p , 0<γ  γ r p γr  γ     , γ p  γ pr  r

,

(14)

where  represents one of the strength parameters, such as c ,  , m , s , and a , p and r represent the peak and residual value of the strength parameters, respectively. γrp is the critical deviatoric plastic strain.

3 Solution processes 12

3.1 Solutions in the elastic zone The stress and displacement solutions in the elastic zone have been presented by Reed. 2

R   =p0   p0   R    , r

 r =p0   p0   R  

2

R  , r

 z =v     r   2vp0  q , u

where v is the Poisson’s ratio, G 

1 R2  p0   R  , 2G r

(15)

(16) (17) (18)

E is the shear modulus, and R is the plastic 2(1  v)

radius. 3.2 Solutions in the plastic zone The plastic region is divided into n concentric annuli. The ith annulus is bounded by two circles with normalized radii of i 1  ri 1 / R and i   ri  / R . At the outer boundary of the plastic region,  0  1 . The stress and strain components can be expressed as  r  0       R     0      ,   q   z  0   

(19)

 r  du / dr     .    u / r 

(20)

It is assumed that the internal support pressure monotonously decreases from

 R to pin after n times, where n is the number of concentric annuli in the plastic zone, and thus the radial stress increment can be determined as 13

 r 

pin   R n

,

(21)

 r i    r i 1   r .

(22)

3.3 Out-of-plane stress as the major principal stress The M–C and generalized H–B failure criteria can be expressed as

 zi 

 mi 1 r(i )    r i    c   si 1     c  

a i 1

,

(23-a)

 z i   N i 1 r i   Yi 1 .

(23-b)

The elastic strain can be expressed using the Hooke’s law as

  

  

1  e   r i   E  r i   v   i    z i    p0  vp0  vq    1  e  i     i   v  r i    z i    p0  vp0  vq   , E  1  e   z i   E  z i   v   i    r i    q  2vp0   

(24)

The circumferential stress can be determined by





  i    p0  vp0  vq   v  r i    z i   Eei 1

(25)

and





    p0  vp0  vq   v  r   z   E  i 1  pi 1 .

(26)

The equilibrium equation is expressed as d r  r     0 , d 

(27)

where  is the radius for the concentric annulus. Equation (28) can be expressed approximately as



 r

i   i 1





 r   0. 

According to Eq. (28), the iterative equation for the radius can be expressed as

14

(28)

 i  

where i  

ri  R

, 



 2   2 

      

r

     r

r

 

(29)

r





,

i 1







1 1 1 i   i 1 ,  r   r i    r i 1 , and      i     i 1 . 2 2 2

The plastic strain can be obtained through the plastic potential function (7) or (9), proposed by Pan and Brown (1996). Based on the plastic flow rule, the plastic strain can be obtained by d p  d

Q . 

(30)

The increments in plastic strain can be expressed based on the M–C failure criterion.   2 z      r   d zp      d , 6 J2  

(31-a)

  2    z   r   dp      d , 6 J2  

(31-b)

  2 r   z         d . d rp   6 J2  

(31-c)

The plastic strains for the generalized H–B failure criterion are determined by solving Eq. (30).  3  2 z      r  1   1 d zp     n   2 z      r   d , 3  c 12 J 2  

(32-a)

 3  2    r   z  1   1 dp     n   2    r   z   d , 3  c 12 J 2  

(32-b)

 3  2 r      z  1   1 d rp     n   2 r      z   d . 3  c 12 J 2  

(32-c)

For the quasi-plane-strain problem of the tunnel, the axial strain is a constant and

15

can be expressed as

 z   ze + zp   0 ,

(33)

 zp   0   ze .

(34)

By combining Eqs. (31) and (33) and Eqs. (32) and (33), the following expressions can be derived,  p e L2     0   z  L  1  L  p      e  3 z 0  r L1

 p e     0   z     p      e  0 z  r

and

f3 

f1 

f2 f1

f3 , f1

(36)

 3 2 r z  1 1  n  2 r z  , L2   3 c 12 J2 

 3 2z  r  1  2z  r  ,  n     3 J 12 c 2  

where L1 

 3 2r  z  1 1  n  2r  z  , L3   3 c 12 J 2 

(35)

 2z  r  , 6 J2

f2 

 2 z r   , 6 J2

and

 2r z    . 6 J2

The elastic strain components can be calculated using Eq. (24). By combining Eqs. (24) and (32), the total strains can be expressed as

  p   e  p e  r   r   r  p e  z   z   z

.

(37)

In the inner plastic zones, using  z =   , the generalized H–B and M–C failure criteria can be expressed as

16

a   mi 1 r i   i1   i    r i    c   si 1      c    a ,  i1  mi 1 r i    si 1   z i    r i    c     c 

 z i   N i 1 r i   Yi 1 .    i   N i 1 r i   Yi 1

(38-a)

(38-b)

The solutions for the stress can be derived by substituting Eq. (19) into Eq. (38). The plastic potential functions can be then expressed as

or

 n 3 3 n J2 J2  Q      I1  c 3 2   Q    n I  3 J  3 n J 2 1   z c 2 2 3 

(39-a)

Q     J 2   I1 .  Q  z   J 2   I1

(39-b)

By combining Eqs. (30) and (39), the plastic flow rule can be expressed as d p  d

Q( ) 

 d

Q( z ) 

.

(40)

By combining Eqs. (38), (40), (34), and (35) or (36), the plastic strain can be obtained and the elastic component can be solved using the Hook’s law. The plastic radius can be then expressed as R

r(i )

 i 

.

(41)

The displacements in the plastic region can be expressed as

ui    i   i  R . 3.4 Out-of-plane stress as the intermediate principal stress The M–C and generalized H–B failure criteria can be expressed as

17

(42)

  i 

 mi 1 r i     r i    c   si 1     c  

a i 1

,

  i   N i 1 r i   Yi 1 .

(43-a) (43-b)

The out-of-plane stress can be obtained using the Hooke’s law.





 z i   v  r  i      i   1  2v  p0  E zp i 1 .

(44)

The increments in circumferential and axial stresses can be expressed as

  i     i     i 1 ,

(45)

 z i    z i    z i 1 .

(46)

Furthermore, the elastic strains should satisfy the Hooke’s law. For the ith annulus, by combining Eqs. (27) and (28), the stress equilibrium equation related to the stress increment can be expressed as

 r i    r i 1 i   i 1

 H        m







H  r i 

 i 



 0

,

(47)



where  i  = i   i 1 / 2 and  i  =  i    i 1 / 2 . r i

c

i 1

 r i  /  c  si 1

 a

i 1





and H  σ r i   = N i - 1 σ r i  +Y i for the  

generalized H–B and M–C failure criteria, respectively. Using Eq. (47), we obtain

i   i 1

  2 H      

2 H  r i    r r i

.

(48)

r

According to Eq. (20), it leads d      r  0. d 

By combining Eqs. (37), (24), and (49), we obtain

18

(49)

 d p  p   rp d e  e   re       d   d   e e 1 v     r = E     r 

.

(50)

By solving Eq. (50), we obtain d p d



 p   p r





d e d



1  v H  r  . E 

(51)

By combining Eqs. (7), (9), (30), (31), (32), and (51), the increment in circumferential plastic strain can be expressed as





 1 ei  1  v H  r i  k1  k3    1   i      pi 1   rpi 1 . (52)   i  E  i  k1   i   i   i  p





For the generalized H–B failure criterion, we obtain  3  2    z   r  1  1   n   2    z   r  k1    c 3  12 J 2  ,

 3  2 r   z     1  1 k3     n   2 r   z     .  3  c 12 J 2 

while for the M–C failure criterion, k1 

 2    z   r    6 J2

and k3 

 2 r   z       6 J2

.

By combining Eqs. (30)–(33), the increments in radial and axial plastic strains can be expressed as

 rpi   pi 

k2 , k1

(53)

 zpi   pi 

k3 , k1

(54)

 3  2 z      r  1  1 k    n   2 z      r  for the generalized H– where 2  c 3  12 J 2 

19

B failure criterion and k2 

 2 z      r    6 J2

for the M–C failure criterion.

The total strains can be then expressed as

 i    i 1  ei   pi    e p  r i    r i 1   r i    r i  .  e p  z i    z i 1   z i    z i 

(55)

The radius and displacement in the plastic region can be obtained using Eqs. (41) and (42).

4 Validations To verify the validity and accuracy of the proposed approach, the results for the proposed approach are compared with those presented by Pan and Brown [10]. The results for the displacements and stresses for the M–C and generalized H–B failure criteria are presented in Table 1 and Fig. 3. The calculation parameters for the generalized H–B failure criterion are obtained from the study by Pan and Brown (1996): p0 = 40 MPa, E = 32 GPa, v = 0.2, pin = 0 MPa, r0 = 1 m, c = 30 MPa, mp = 8.78, sp = 0.189, mr = 5.14, sr = 0.082, ap = 0.5, and ar = 0.5. The calculation parameters for the M–C failure criterion are determined based on the study by Pan and Brown: p0 = 40 MPa, E = 32 GPa, v = 0.2, pin = 0 MPa, r0 = 1 m, Np = 3.2, Yp = 30 MPa, Nr = 2.2, and Yr =10 MPa. Table 1 Plastic radii and inner wall displacements for different dilation parameters (H–B) Dilation

Plastic radius

Inner wall displacement

20

parameter

n

R (m)

u (cm)

Pan and Brown’s

Proposed

Pan and Brown’s

Proposed

solution

approach

solution

approach

0.1

1.65

1.65

0.70

0.68

1.2

1.65

1.65

0.89

0.89

2.3

1.65

1.65

1.20

1.25

3.5

1.65

1.65

2.17

2.29

Table 1 shows that the results obtained using the proposed approach are very similar to those obtained by Pan and Brown(1996) when the out-of-plane stress is 45 MPa (i.e., q = 45 MPa) for the carious dilation parameters. The difference between the inner wall displacement results obtained by the proposed approach and those obtained by Pan and Brown(1996) is smaller than (2.29−2.17)/2.29 × 100% = 5.2%.

21

Fig. 3. Stress verification. (a), (b) Stresses for the H–B and M–C criteria, respectively. Figures 3(a) and 3(b) show the stress distributions of the surrounding rock obtained using the proposed approach and Pan and Brown’s solution, respectively. For the M–C and generalized H–B failure criteria, the stresses and plastic radius of the proposed approach are similar to the Pan and Brown’s solutions. Moreover, the results obtained by the proposed approach are in good agreement with those obtained using the numerical method. The small difference in axial stress in the plastic zone might be attributed to the different assumptions regarding the axial stress and initial axial plastic strain. When the out-of-plane stress and axial strain are considered, the stress distributions are in good agreement with each other, which demonstrates the validity of the proposed method.

5 Influences of the in situ stress on the stress and displacement 5.1 Out-of-plane stress as the major principal stress To illustrate the influence of the out-of-plane stress as the major principal stress,

22

the stress and displacement of the surrounding rock with various out-of-plane stresses are calculated based on the parameters presented by Wang et al. (2012) and Pan and Brown (1996). Considering the softening properties of the rock mass, the results obtained with different critical softening parameters (γp) of 0, 0.008, 0.012, 0.016, 0.02, and 100 (any large value) are analyzed.

Fig. 4. Distributions of the stress and displacement with different critical softening parameters (γp) (H–B) of (a) 0.008 and (b) 0.016.

23

Fig. 5. Distributions of the stress and displacement with different critical softening parameters (γp) (M–C) of (a) 0.008; (b) 0.016; (c) 0, 0.008, 0.012, 0.02, and 100. Figures 4 and 5 show the effects of the out-of-plane stress on the distributions of

24

the stress and displacement when the out-of-plane stress is the major principal stress with different critical softening parameters. The entire plastic region consists of two parts, the outer plastic zone where  z    and inner plastic zone where  z    . At the interface between these two zones, Rs < Rf and  z    . The outer plastic zone is smaller than the softening zone (i.e., Rs < Rf). Moreover, Fig. 5 shows that the effects of the critical softening parameters on the distributions of the stresses and displacements are significant. For the M–C failure criterion, a larger softening parameter corresponds to smaller plastic zone and displacement of the surrounding rock. With the increase in critical softening parameter, the size of the residual region decreases, whereas that of the softening region increases. These results are similar to those presented by Wang et al. (2012).

25

Fig. 6. Stress distributions with different dilation parameters. (a)    (M–C), (b)

   (M–C), (c) n = 0.1 (H–B), and (d) n = 2 (H–B). 26

Fig. 7. Tunnel wall displacement as a function of the dilation parameter. (a) M–C, (b) H–B. Figures 6 and 7 demonstrate the influences of the dilation parameter on the stress and displacement of the surrounding rock when the out-of-plane stress is the major principal stress. Figure 6 shows that the effect of the dilation on the stress is insignificant; however, its effect on the plastic region is significant. With the increase in dilation parameter, the size of the softening region decreases, whereas that of the residual region increases. The strength of the surrounding rock decreases with the 27

increase in dilation parameter. For example, the radius of the residual region increases from 1.1 to 1.54 m and the entire plastic radius increases from 1.53 to 1.63 m when the dilation parameter based on the M–C failure criterion increases from 0 to 0.21. The residual radius increases from 1.34 to 1.64 m and the entire plastic radius increases from 2.18 to 2.27 m when the dilation parameter based on the generalized H–B failure criterion increases from 0.1 to 0.20. Figure 7 demonstrates the relationship between the displacement of the tunnel wall (Ur) and dilation parameter. It can be concluded that the displacement of the surrounding rock increases slightly with the dilation parameter. When the dilation parameter increases to a certain value, the change in dilation parameter has a significant influence on the displacement of the tunnel wall. For example, when the dilation parameter based on the generalized H–B failure criterion increases from 0 to 0.20, the displacement of the tunnel wall increases slightly with the dilation parameter. However, when the dilation parameter is larger than 2.0, the displacement of the tunnel wall increases considerably with the dilation parameter. Therefore, a proper consideration of the dilation parameter is important. 5.2 Out-of-plane stress as the intermediate principal stress When the out-of-plane stress is considered as the intermediate principal stress, the corresponding stress distributions and displacement of the surrounding rock are shown in Figs. 8, 9, 10, and 11.

28

Fig. 8. Stress distribution based on the H–B criterion. (a) n = 0.1, (b) n = 2.5.

29

Fig. 9. Stress distribution based on the M–C criterion. (a) α = 0, (b) α = 0.21. Figures 8 and 9 show the stress distributions when the out-of-plane stress is considered as the intermediate principal stress for n = 0.1 and 2.5 and  = 0 and 0.21, respectively. Only one softening zone in the plastic region is observed when n = 0.1. However, when n is 2.5, residual and softening zones are observed. When n = 2.5, the out-of-plane stress around the tunnel wall is more equivalent to the circumferential stress than that when n = 0.1. For example, the plastic radius equal to the softening 30

radius is approximately 1.5 m when n = 0.1. However, when n = 2.5, the residual radius is approximately 1.2 m and the plastic radius is approximately 1.5 m. As shown in Fig. 9, similar trends can be observed for the results obtained using the proposed approach based on the M–C failure criterion.

Fig. 10. Displacement distributions for different dilation parameters based on the (a) H–B and (b) M–C criteria. Figure 10 shows the displacement distributions around the tunnel for different dilation parameters. As shown in Fig. 10(a), based on the H–B failure criterion, the displacement gradually increases with the dilation parameter n in the range of 0.1 to

31

2.5. The displacements of the tunnel inner wall are 0.42, 0.46, 0.52, 0.58, 0.67, and 0.79 cm at n of 0.1, 0.5, 1.5, 2.0, and 2.5, respectively, whereas the plastic radii are approximately 1.5 m. Therefore, the effect of the dilation parameters on the displacements of the surrounding rock is significant, while that on the plastic region is insignificant. As shown in Fig. 10(b), similar trends can be observed for the M–C failure criterion. The displacement gradually increases with the dilation parameter  in the range of 0 to 0.21.

Fig. 11. Tunnel wall displacements as a function of the dilation parameter based on the (a) H–B and (b) M–C criteria.

32

The relationship between the displacement at the tunnel wall and dilation parameter is shown in Fig. 11. A larger dilation parameter usually leads to a larger displacement in the plastic region. When the dilation parameter is larger than 2.0, a considerably increased displacement can be observed at the tunnel wall. The distributions of the stress, displacement, and ground reaction curves obtained considering the out-of-plane stress under different critical values of the strain-softening parameter (  r = 0, 0.008, 0.012, 0.02, and 100) are presented in Figs. p

12, 13, 14, 15, and 16. The calculation parameters are presented by Pan and Brown (1996) except in the case of ap = 0.51, ar = 0.55, and n = 2.0.

33

Fig. 12. (a) Stress and (b) displacement distributions for the elastic-brittle–plastic model (H–B).

(a)

(b)

Fig. 13. (a) Stress and (b) displacement distributions for γp = 0.008 (H–B).

34

Fig. 14. (a) Stress and (b) displacement distributions for γp = 0.012 (H–B).

35

Fig. 15. (a) Stress and (b) displacement distributions for the elastic–plastic model with γp = 100 (H–B).

36

37

Fig. 16. (a), (c) Stress and (b), (d) displacement distributions with different softening parameters based on the H–B and M–C failure criteria, respectively. Figure 12 shows the distributions of the stress and displacement for the elastic-brittle–plastic model when γp = 0. Only a plastic zone exists around the tunnel wall. Apparent reductions in circumferential and out-of-plane stresses are observed at the elastic–plastic interface. Figures 13 and 14 show that the plastic region consists of softening and residual zones when the strain softening parameter is considered (γp = 0.008). In such cases, a larger critical softening parameter generally leads to a smaller residual zone and larger softening zone. For example, when γp = 0.008, the residual radius is approximately 1.2 m while that at γp = 0.012 is approximately 1.1 m. Figure 15 shows the stress and displacement distributions based on the elastic–plastic model when γp =100. Only a plastic zone exists around the tunnel wall and the circumferential and out-of-plane stresses at the elastic–plastic interface are continuous. According to Fig. 16, similar

38

rules can be observed for the M–C and H–B failure criteria. The stresses in the plastic region gradually increase with the softening parameter γp, while the displacements and plastic radius decrease with the increase in γp.

6 Conclusions A new simple approach for the theoretical solution for the excavation of a circular tunnel in a strain-softening rock mass considering the out-of-plane stress was proposed. The plastic axial strain and deformation-dependent intermediate principal stress were employed based on the assumption of quasi-plane strain. According to the obtained numerical results, the following conclusions can be summarized: (1) Theoretical solutions could be obtained for the intermediate principal stress and axial strain based on the proposed approach. The influences of the out-of-plane stress and plastic strain on the mechanical analysis of the tunneling could be effectively analyzed. The out-of-plane plastic strain had an important role in the evaluation of the out-of-plane stress σz and could not be zero. (2) Based on the theoretical solutions, the stress in the plastic region of the surrounding rock and displacement in the tunnel wall could be obtained by considering the out-of-plane stress as either the major or intermediate principal stress. (3) Compared with the solutions presented by Pan and Brown (1996), the theoretical solutions proposed in this study are effective and accurate. The characteristics of the stress and displacement distributions of the surrounding rock obtained in this study are consistent with those obtained by Pan and Brown (1996). (4) The influences of the out-of-plane stress and dilation parameters on the stress 39

in the plastic region of the surrounding rock and that of the displacement on the tunnel wall were discussed. When the critical softening parameter was in the range of 0 to 100 (γp=0.008), the plastic region consisted of softening and residual zones. When the dilation parameter was larger than 2.5, a considerably increased stress was observed in the plastic region.

Acknowledgements This work was supported by the National Basic Research Program of China (“973” Project) (Grant No. 2013CB036004) and the National Natural Science Foundation of China (Grant No. 51208523).

Conflict of interest The authors declared that they have no conflicts of interest to this work.

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