Elasto-plastic analysis of a circular opening in rock mass with confining stress-dependent strain-softening behaviour

Tunnelling and Underground Space Technology 50 (2015) 94–108

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

Elasto-plastic analysis of a circular opening in rock mass with confining stress-dependent strain-softening behaviour Lan Cui, Jun-Jie Zheng ⇑, Rong-Jun Zhang, You-Kou Dong Institute of Geotechnical and Underground Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e

i n f o

Article history: Received 5 January 2014 Received in revised form 23 June 2015 Accepted 4 July 2015

Keywords: Circular opening Strain-softening Variable confining stress Elasto-plastic solution Hoek–Brown criterion

a b s t r a c t In strain-softening rock mass, it is widely accepted that the confining stress in the plastic zone around a circular opening varies with the radial distance. The variable confining stress gives rise to a variable critical plastic softening parameter (g⁄). This paper aims to take the variable g⁄ into account in the development of elasto-plastic solutions for stress and strain states around a circular opening. First, a variable confining stress model (VCSM) is recommended to account for the effect of the variable g⁄; a criterion is presented for judging whether or not the rock mass transfers from the plastic softening state to the residual state. Then, a new numerical procedure for the implementation of VCSM is proposed on the basis of Hoek–Brown failure criterion and the non-associated flow rule. In the proposed procedure, the plastic softening and residual zones are divided into a set of concentric rings by an assigned radial stress increment. The increments of stress and strain for each ring can be calculated in a successive manner by the finite difference method. Finally, by using the proposed procedure, a series of parametric studies is conducted to compare VCSM and existing constant confining stress models (CCSM), and also to investigate the variation of the stress components, strain-softening parameter, and strength parameters in the plastic softening zone. The results indicate that CCSM tends to underestimate the deformation of surrounding rock mass with poor quality. The effect of the critical plastic softening parameter on the stability of a circular opening in strain-softening rock mass includes two different aspects: one is to govern the strength parameters of the plastic zone, the other is to control the radii of the plastic softening and residual zones. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Elasto-plastic analysis of stress and displacement around a circular opening in isotropic rock mass is a fundamental problem in tunnel engineering. It is of important value in the evaluation of stability and the design of support structure. According to previous studies (Carranza-Torres and Fairhurst, 2000, 1999; Park and Kim, 2006; Sharan, 2003, 2005; Serrano et al., 2011; Brown et al., 1983; Carranza-Torres, 1998; Wang, 1996; Alonso et al., 2003; Guan et al., 2007; Lee and Pietruszczak, 2008; Wang et al., 2010, 2011; Zhang et al., 2012; Sharan, 2008; Park et al., 2008), the elasto-plastic solutions were generally developed via analytical and semi-analytical methods based on the axial symmetry or plane strain assumption. A majority of these available solutions was based on the elastic-perfectly plastic model or elastic–brittle– plastic model for the sake of simplicity. However, it is commonly observed in field and laboratory (Hoek et al., 2002; Alejano and Alonso, 2005; Detournay, 1986; Yuan and Harrison, 2007; Alonso ⇑ Corresponding author. E-mail address: [email protected] (J.-J. Zheng). http://dx.doi.org/10.1016/j.tust.2015.07.001 0886-7798/Ó 2015 Elsevier Ltd. All rights reserved.

et al., 2008; Hoek and Brown, 1997; Cai et al., 2004, 2007; Medhurst, 1996; Pethukov and Linkov, 1979; Hoek and Diederichs, 2006) that rock mass with average quality tends to show strain-softening behaviour during the post-failure stage. Therefore, in recent years, effort was made by a number of researchers to account for the strain-softening effect. For instance, Brown et al. (1983) and Carranza-Torres (1998) presented different numerical procedures for the analysis of stress and displacement of a circular opening in strain-softening material. Subsequently, authors (Wang, 1996; Alonso et al., 2003; Guan et al., 2007; Lee and Pietruszczak, 2008; Wang et al., 2010, 2011; Zhang et al., 2012; Varas et al., 2005) furthered the methods in Brown et al. (1983) and Carranza-Torres (1998), in an effort to gain more accurate and convenient solutions. Typical examples include Alonso’s method (Alonso et al., 2003) and Lee and Pietruszczak’s method (Lee and Pietruszczak, 2008). The former is commonly recognised to be rigorous though somewhat complicated for practical application. The latter obtained the numerical solution by dividing the potential plastic zone into a finite number of concentric rings. The critical plastic softening parameter g⁄ (as defined in Fig. 1), controlling the transition from plastic softening to residual stage, is

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95

List of symbols

g⁄

critical plastic softening parameter variable confining stress model constant confining stress model r3;crit confining stress at the elasto-plastic boundary rpeak peak strength in the stress-strain curve with a certain 1 confining stress rr radial stress rh tangential stress r3 confining stress; minor principal stress r1 major principal stress r0 hydrostatic stress rr2 radial stress at the elasto-plastic boundary rh2 tangential stress at elasto-plastic boundary rr1 radial stress at the plastic softening-residual boundary rh1 tangential stress at plastic softening-residual boundary R0 radius of the opening Rp radius of the plastic softening zone Rr radius of the plastic residual zone pi support pressure pic;2 support pressure for the occurrence of plastic softening zone pic;1 support pressure for the occurrence of plastic residual zone f failure criterion g plastic potential rij stress tensor g plastic softening parameter rci uniaxial compression strength of rock mass in the intact state mb ; s; a strength parameters of H–B failure criterion ep1 major principal plastic strain ep3 minor principal plastic strain x one of strength parameters (mb , s or a) of H–B failure criterion E Young’s modulus Kw dilatancy coefficient of rock mass w dilatancy angle mpeak ; speak ; apeak peak value of mb , s or a in the elastic zone b VCSM CCSM

important to a rational solution, since it influences the strength parameters and dilatancy (Carranza-Torres and Fairhurst, 2000, 1999; Park and Kim, 2006; Sharan, 2003, 2005; Serrano et al., 2011; Brown et al., 1983; Carranza-Torres, 1998; Wang, 1996; Alonso et al., 2003; Guan et al., 2007; Lee and Pietruszczak, 2008; Wang et al., 2010, 2011; Zhang et al., 2012; Varas et al., 2005). Most of existing references regarded the critical plastic softening parameter as constant. This is not true when the variable confining stress in the plastic softening zone is considered. It was suggested by Alejano et al. (2010, 2012) that the mechanical response of strain-softening rock mass should be confining-stress-dependent. Unfortunately, in the source reference, Alejano et al. regarded r3;crit =2 (referring to Fig. 2) as the confining stress in the whole plastic zone for simplicity, regardless of the variability of confining stress in the plastic zone. This still leads to a constant g⁄. It was also observed in laboratory tests that rock mechanical parameters such as the critical plastic softening parameter (Alejano et al., 2010, 2012), shear strength and dilatancy (Alejano and Alonso, 2005; Detournay, 1986; Yuan and Harrison, 2007; Medhurst, 1996) were heavily dependent on confining stress. The effect of confining stress on g⁄ and rpeak (peak strength 1 of rock mass) can be schematically depicted in Fig. 3. The point is that the critical plastic softening parameter g⁄ increases with the

res res mres residual value of mb , s or a in the plastic residual b ;s ;a zone M drop modulus of post peak stage GSI Geological Strength Index GSIp peak value of Geological Strength Index, representing rock mass strength in the elastic zone u radial displacement of the rock mass u0 radial displacement of the rock mass at the circular opening r radial distance to the centre of the opening. rer radial stress in the elastic zone reh tangential stress in the elastic zone ue radial displacement in the elastic zone r ði1Þ ; r ðiÞ radii of inner and outer boundaries of the ith annulus rrð0Þ initial value of radial stress, equal to rr2 rhð0Þ initial value of tangential stress, equal to rh2 Drr constant radial stress increment DrhðiÞ tangential stress increment rrði1Þ ; rrðiÞ radial stresses at r ¼ rði1Þ and r ¼ rðiÞ gðiÞ critical plastic softening parameter at r ¼ r ðiÞ gðiÞ plastic softening parameter at r ¼ rðiÞ erðiÞ ; erði1Þ radial strains at r ¼ r ði1Þ and r ¼ r ðiÞ ehðiÞ ; ehði1Þ tangential strains at r ¼ r ði1Þ and r ¼ r ðiÞ DephðiÞ tangential plastic strain increment DeprðiÞ radial plastic strain increment DeehðiÞ tangential elastic strain increment DeerðiÞ radial elastic strain increment DgðiÞ increment of plastic softening parameter r0rðiÞ average value of rrði1Þ and rrðiÞ uðiÞ radial displacement at r ¼ r ðiÞ DuðiÞ increment of radial displacement at r ¼ r ðiÞ j number of the annulus immediately outside the plastic softening-residual boundary gaver value of g⁄ solved by Eq. (9) when considering the confining stress as rr2 =2

Fig. 1. Schematic graph for stress–strain curve in plastic softening zone and residual zone [note: eph and erh stand for the tangential and radial plastic strains].

increase in confining stress. In other words, g⁄ is not a constant in the plastic zone. It turns out that the strain–stress curve in the plastic softening zone varies from point to point due to the variable

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Fig. 4. Analysis model of rock mass after excavation. Fig. 2. Calculation of the average value of the confining stress (Alejano et al., 2010).

confining stress. Therefore, the constant confining stress models (referred to as CCSM from hereon), such as Alejano’s model (Alejano et al., 2010, 2012), seem to be unreasonable in coping with this issue. The objective of this paper is to develop a new numerical procedure to account for the variable confining stress in the strain-softening model. To be specific, this paper mainly covers four parts: (1) to propose a variable confining stress model (VCSM), and a modified criterion to judge whether the rock mass resides into the plastic residual state; (2) to present the implementation procedure for VCSM based on the finite difference method (FDM); (3) to validate the proposed VCSM and its implementation procedure via several examples; and (4) to conduct a series of parametric studies to compare CCSM and VCSM. 2. Problem description 2.1. Basic assumptions Following assumptions are to be employed in the development of VCSM. (1) The opening is circular. (2) Rock mass is isotropic, continuous, infinite and initially elastic; and it shows strain-softening behaviour obeying Hoek– Brown (H–B) criterion. (3) The opening is excavated in a hydrostatic stress field. The stress distribution around the opening is axisymmetric.

(4) Plane strain condition is postulated. rr and rh represent the minor principal stress r3 (i.e. the confining stress) and the major principal stress r1 , respectively.

2.2. Excavation problem In accordance with above assumptions, the excavation problem can be described in Fig. 4. A hydrostatic stress field r0 exists prior to the excavation. The radius of the opening is R0 , rr2 and rh2 are the radial and tangential stresses at elasto-plastic boundary. rr1 and rh1 are the radial and tangential stresses at plastic softening-residual boundary. The radii of the plastic softening and residual zones are denoted by Rp and Rr , respectively. A support pressure pi is uniformly distributed along the excavation boundary. As pi gradually decreases, the deformation of the surrounding rock develops. pic;2 and pic;1 represent the support pressures for occurrences of the plastic softening and residual zones, respectively. When pi is lower than pic;2 , the plastic softening zone occurs. When pi is lower than pic;1 (pic;1 < pic;2 ), the plastic residual zone occurs. Theoretically, pic;2 and pic;1 are equal to rr2 and rr1 , this will be validated further below.

3. Alejano’s model Although Alejano’s model (Alejano et al., 2010, 2012) seems to be unreasonable in solving the problem depicted in Fig. 4, some arguments in their studies are heuristic and of practical

Fig. 3. Schematic graph for the variation of the g versus r3.

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importance in the development of VCSM. Hence, Alejano’s model is first described in this section for reference. 3.1. H–B failure criterion According to the theory of plasticity (Hill, 1950; Kaliszky, 1989), the deformation process is characterised by a failure criterion f and a plastic potential g. f and g depend on the stress tensor rij and the plastic softening parameter g. The failure criterion can be defined as follows:

f ðrij ; gÞ ¼ 0

ð1Þ

The latest H–B failure criterion (Hoek et al., 2002) is written as:



r1 ¼ r3 þ rci mb

r3 þs rci

a ð2Þ

Fig. 5. Variation of the strength parameter x in plastic zone versus g.

in which rci is the uniaxial compression strength of rock mass in the intact state; mb , s and a are the strength parameters of rock mass. For the strain-softening rock mass surrounding a circular opening, H–B failure criterion can be written as:

 aðgÞ rr f ðrh ; rr ; gÞ ¼ rh  rr  rci mb ðgÞ þ sðgÞ ¼0

rci

ð3Þ

where mb , s and a are functions of the plastic softening parameter g. 3.2. Plastic softening parameter According to Alonso et al. (2003), the softening parameter in the case of plane strain (e2 ¼ ep2 ¼ 0) is often assumed to be the difference between the major and minor principal plastic strains, i.e.:

g ¼ ep1  ep3

ð4Þ

Referring to the assumption of axial symmetric condition, Eq. (4) is rewritten as:

g ¼ eph  epr

ð5Þ

Essentially, g is the plastic shear strain. Fig. 5 shows the relation between g and the strength parameters mb , s or a which can be expressed as:

(

xðgÞ ¼

xpeak  ðxpeak  xres Þ gg ; 0 < g < g xres ; g P g

ð6Þ

where x represents any of the strength parameters (mb , s or a), and g* is the critical plastic softening parameter shown in Fig. 5. It can be seen from Eq. (6) that during the plastic softening stage, the strength parameters reduce linearly with the increase in g, and beyond the critical value g* (i.e. in the plastic residual stage), they remain unchanged. By studying the experimental and numerical data (Hoek and Brown, 1997; Duncan Fama et al., 1995; Medhurst and Brown, 1998) concerning different rock masses, coal pillars and coal samples, Alonso et al. (2008) argued that g⁄ may vary between 0.01 and 0.001. From a practical standpoint, Cai et al. (2004, 2007) pointed out that g⁄ is in a range of about 5–10 times the strain corresponding to the peak strength. Nevertheless, this range may still be too wide to derive practically applicable solutions for the deformation of surrounding rock mass. With this consideration, Alejano et al. (2010) proposed a geometrical graph depicting the stress–strain relationship for a strain-softening rock mass (shown in Fig. 6). By Fig. 6, g⁄ can be estimated as:

1 1  ð1 þ K w Þ g ¼ rpeak  rres þ 1 1 E M 

ð7Þ

Fig. 6. Geometrical graph for the estimation of g⁄ (Alejano et al., 2010).

where E is the Young’s modulus; M is the drop modulus of the post peak failure behaviour; rpeak and rres 1 are the peak and residual val1 ues of the rock mass strength; K w is the dilatancy coefficient of rock mass. Note that Eq. (7) is defined in terms of a standard triaxial test in which the confining stress is applied to a sample peripherally (Alejano et al., 2010). K w is equal to

1þsin w 2ð1sin wÞ

(Alejano and Alonso,

2005). During the plastic softening and residual stages, Eq. (3) can be rewritten to the following equations:



rpeak ¼ r3 þ rci mpeak r3 =rci þ speak 1 b 

res res rres 1 ¼ r3 þ rci mb r3 =rci þ s

apeak

ð8aÞ

ares

ð8bÞ

where mpeak , speak , and apeak are the peak values of mb , s or a; and b res res mb , s , and ares are the residual values of mb , s or a. Previous laboratory and field test results indicated that the drop modulus M depends heavily on rock mass quality (Hoek and Brown, 1997; Cai et al., 2007) and confining stress (Alejano and Alonso, 2005; Hoek and Diederichs, 2006). Therefore, Alonso et al. (2008) defined M as follows:

M¼

8  > r3 0:00768GSIp > pffiffiffiffiffiffiffi > E  ½0:0046e  < 2

speak rci

1 þ 0:05

 1 > > p r3 > : E  ½0:0046e0:00768GSI  pffiffiffiffiffiffiffi speak rci

r3 pffiffiffiffiffiffiffi

6 0:1

r3 pffiffiffiffiffiffiffi

P 0:1

speak rci

speak rci

ð9Þ

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where GSIp is the peak value of GSI, representing the strength of rock mass in the elastic zone. Incorporating Eqs. (8) and (9) into Eq. (7), one obtains:

8   pffiffiffiffiffiffiffi > ðr3 þ0:1 speak rci Þ r3 > pffiffiffiffiffiffiffi > 6 0:1 < F  1 þ 0:0092e0:0768GSIp pffiffiffi sp rci speak rci   g ¼ > r3 pffiffiffiffiffiffiffi r3 > > pffiffiffiffiffiffiffi P 0:1 :F  1þ 0:0768GSIp peak peak 0:0046e

where

s

rci

s

ð10Þ

rci

  apeak apeak r3 r3 peak peak F ¼ rEci ð1 þ K w Þ ðmpeak Þ  ðmpeak Þ . b b rci þ s rci þ s

Apparently, g highly relies on the confining stress

r3 . Fig. 8. Schematic graph of stress–strain relationships for different values of r3.

4. Development of VCSM 4.1. Proposal on VCSM In Alejano’s model (Alejano et al., 2010), a value of r3;crit =2 (r3;crit is the confining stress at the elasto-plastic boundary) is selected to represent the confining stress r3 of the whole plastic zone. Apparently, the assumed confining stress path illustrated in Fig. 2 cannot reflect the reality. First, the presence of pi influences the confining stress path (at least the confining stress at the opening surface – 0), but this was not considered in Fig. 2. Second, due to the different mechanical behaviours in the plastic softening and residual zones, the variation path of the confining stress should be of different trends in the two zones, thus the path shown in Fig. 2 was overly simplified by Alejano et al. (2010). Fig. 7 depicts the theoretical confining stress path in the plastic softening and residual zones. When the stress state of rock mass transfers from the plastic softening stage to plastic residual stage, the major and minor principal stresses r1 and r3 gradually fall from rh2 and rr2 to rh1 and rr1 , respectively. While in the plastic residual zone, confining stress r3 drops from rr1 at the plastic softening-residual boundary to pi at the opening surface. Fig. 8 shows the schematic curves of stress–strain relationships for different confining stresses obtained by Eq. (10). The results indicate that in the plastic softening zone, g is variable and the strain– stress relationship in plastic zone is no longer unique due to the variable confining stress. This means Alejano’s model (Alejano et al., 2010), in which a constant confining stress in the plastic zone was adopted, seems to be overly simplified. Therefore, a variable confining stress model (VCSM) is proposed in this paper to obtain an elasto-plastic solution. VCSM strictly follows the confining stress path shown in Fig. 7 and the diverse stress–strain relationships shown in Fig. 8.

Fig. 9. Dividing method for the variable confining stress model.

Fig. 10. Judging criterion for entering the plastic residual zone.

4.2. Governing equations For the case of plane strain, the equilibrium equation is

Fig. 7. Variation of r3.

@ rr rr  rh þ ¼0 @r r

ð11Þ

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In terms of small strain case, the displacement compatibility is

er ¼

du ; dr

eh ¼

u r

ð12Þ

where u is the radial displacement of the rock mass, and r is the radial distance to the centre of the opening. 4.3. Stress and deformation in the elastic zone In the elastic zone, the radial stress rer , tangential stress reh , and displacement ue , can be obtained in accordance with Eqs. (11) and (12) (Hill, 1950), i.e.:

rer ¼ r0  ðr0  rr2 Þ

 2

r ¼ r0 þ ðr0  rr2 Þ e h

Rp r

 2

ð13Þ

Rp r

2

ue ¼

ð1 þ lÞðp0  rr2 Þ Rp  E r

ð14Þ

ter and softening parameter at ith annulus. With the decrease in distance to the opening surface, gðiÞ drops but gðiÞ increases as the plastic deformation develops. Provided that the plastic residual zone exists (namely, pi < pic;1 ), gðiÞ gradually increases to a value equal to gðiÞ at a certain position near the opening surface. As g becomes variable, the curves for strength parameter (mb , s or a) versus softening parameter g are nonlinear. If the unloading path of rock mass is considered, there should exist the increment of the elastic strain. The plastic strain is then expressed as:



erðiÞ ehðiÞ







(

DeerðiÞ erði1Þ ¼ þ DeehðiÞ ehði1Þ

)

( þ

DeprðiÞ

)

DephðiÞ

ð17Þ

where erðiÞ , ehðiÞ , erði1Þ and ehði1Þ are the radial and tangential strains at r ðiÞ and r ði1Þ , respectively; DephðiÞ and DeprðiÞ are the tangential and radial plastic strain increments; and DeehðiÞ and DeerðiÞ are the tangen-

Substituting Eq. (13) into Eq. (8a), one obtains:

 apeak rci mpeak rr2 =rci þ speak þ 2rr2  2r0 ¼ 0 b

A criterion to judge whether or not the rock mass transfers from the plastic softening state to the residual state is proposed in Fig. 10. gðiÞ and gðiÞ represent the critical plastic softening parame-

ð15Þ

where rr2 can be solved by use of the Newton–Raphson method. However, the stress and displacement within the elastic zone (not including the elasto-plastic boundary) remain unsolved because Rp is still unknown. 4.4. Stress and deformation in the plastic softening and residual zones Closed-form solutions are derived for both elastic-perfectly plastic and elastic–brittle–plastic rock masses following the linear M–C (Park and Kim, 2006; Reed, 1986; Ogawa and Lo, 1987) or the non-linear H–B failure criterion (Sharan, 2003, 2005; Carranza-Torres and Fairhurst, 1999; Detournay, 1986). In contrast, it seems to be complicated to develop a closed-form solution for the strain-softening rock mass obeying the H–B criterion due to the nonlinear failure envelope. Moreover, to further consider the variable confining stress r3 in the plastic zone will make this task fairly difficult The finite difference method (FDM), as proposed by Brown et al. (1983), is adopted in this study to solve the problem in a numerical way, rather than in a closed-form. The plastic zone is divided into a set of concentric annuli. As shown in Fig. 9, rði1Þ and r ðiÞ are the radii of the inner and outer boundaries of the ith annulus. At the outer boundary of the plastic zone, rrð0Þ and rhð0Þ are equal to rr2 and rh2 at the elasto-plastic boundary. As proposed in Lee and Pietruszczak (2008), a constant radial stress increment is assumed for each annulus, i.e.:

tial and radial elastic strain increments. According to Hooke’s law, the elastic strain increments can be correlated to the stress increments (Lee and Pietruszczak, 2008), i.e.:

(

DeerðiÞ DeehðiÞ

) ¼

  DrrðiÞ l 1þl 1l E DrhðiÞ l 1  l

ð18Þ

Based on Eq. (5), the increment of plastic softening parameter DgðiÞ at the ith annulus can be described as:

DgðiÞ ¼ DephðiÞ  DeprðiÞ

ð19Þ

In accordance with non-associated flow rule, the following equation is formulated

DeprðiÞ ¼ K w DephðiÞ

ð20Þ

w 1þsin w where K w is the dilatancy coefficient, K w ¼ 1þsin or 2ð1sin , and w is 1sin w wÞ

the dilatancy angle. Integrating Eq. (17) into Eq. (20), there is

erðiÞ þ K w ehðiÞ ¼ erði1Þ þ K w ehði1Þ þ DeerðiÞ þ K w DeehðiÞ

ð21Þ

In each annulus, Eq. (11) can be transformed into the differential form as follows (Lee and Pietruszczak, 2008):

rrðiÞ  rrði1Þ r ðiÞ  r ði1Þ in which,



Hðr0rðiÞ ; gði1Þ Þ ðr ðiÞ þ r ði1Þ Þ=2

¼0

ð22Þ

r0rðiÞ ¼ ðrrðiÞ þ rrði1Þ Þ=2 and

8  . aði1Þ > ; for the plastic softening zone < rci mb ði  1Þr0rðiÞ rci þ sði1Þ Hðr0rðiÞ ; gði1Þ Þ ¼ res  .  a > res : rci mres r0 ; for the plastic residual zone rðiÞ rci þ s b

( p r

r2

i

Drr ¼

n

rrðiÞ  rrði1Þ

ð16Þ

where rrði1Þ and rrðiÞ denote the radial stresses at r ¼ r ði1Þ and r ¼ r ðiÞ (shown in Fig. 9), respectively; n is the number of concentric annuli.

Incorporating Eq. (14) into Eq. (20), the relation between rðiÞ and rði1Þ can be derived as

r ðiÞ rði1Þ

¼

2Hðr0rðiÞ ; gði1Þ Þ þ Drr 2Hðr0rðiÞ ; gði1Þ Þ  Drr

ð23Þ

In order to solve the strain components, Eq. (12) is rewritten as:

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Table 1 Characterization of analysis conditions A1–E1.

A1 B1 C1 D1 E1

R0 (m)

GSIp

GSIr

r0 (MPa)

rci (MPa)

mpeak

speak

mres

sres

w (deg)

E (GPa)

l

3 3 3 3 3

75 60 50 40 25

40 35 30 27 18

15 15 15 15 15

75 75 75 75 75

4.09 2.379 1.677 1.173 0.6294

0.0622 0.0117 0.0039 0.0013 0.0002

1.173 0.981 0.821 0.737 0.3681

0.0013 0.0007 0.0004 0.0003 0.0001

9.42 5.75 3.81 1.72 0

36.51 15.4 8.66 4.87 2.053

0.25 0.25 0.25 0.25 0.25

Fig. 11. Relations amongst Rr, R0 and Rp calculated for analysis conditions A1–E1: (a) Rr versus Rp and (b) R0 versus Rp.

res @ rr rci ðmres b rr =rci þ s Þ ¼ @r r

ares

ð26Þ

Two boundary conditions for Eq. (26) are: (1) when r ¼ R0 , rr ¼ pi ; and (2) when r ¼ Rr , rr ¼ rr1 . Hence, the following equation can be derived from Eq. (26):

Rr ¼ R0 exp

n h

res rr1 mres b =rci s

mres ð1  ares Þ

1ares

o

 res i. res 1a  pi mres b =rci þ s ð27Þ

4.5. Relations among Rp , Rr and R0

Fig. 12. Relation between Rp and rr1 calculated for analysis conditions A1 - E1.

erðiÞ

DuðiÞ uðiÞ ¼ ; ehðiÞ ¼ Dr ðiÞ r ðiÞ

ð24Þ

Eq. (27) provides a relation between Rr , and rr1 . However, the value of Rr is still unknown. Therefore, the internal relations among Rp , Rr and other given parameters such as R0 must be ascertained in advance. Essentially, Eq. (23) can be rewritten into:

  0 j Rr Y 2H rrðiÞ ; gði1Þ þ Drr   ¼ Rp i¼1 2H r0 ; g  Drr rðiÞ

By combining Eqs. (18), (21) and (24), the radial displacement uðiÞ can be expressed as:

uðiÞ ¼

C ði1Þ r ðiÞ ðrðiÞ  r ði1Þ Þ þ uði1Þ r ðiÞ r ðiÞ þ K w ðr ðiÞ  r ði1Þ Þ

where

uði1Þ

is

the

radial

ð25Þ displacement

at

r ¼ rði1Þ ,

C ði1Þ ¼ Aði1Þ þ Bði1Þ , Aði1Þ ¼ erði1Þ þ K w ehði1Þ , Bði1Þ ¼ ð1þE mÞ fDrrðiÞ ð1  m  K w mÞ þ ½rhði1Þ þ rrðiÞ þ Dði1Þ ðK w  K w m  mÞg, and

(

rci ðmbði1Þ rrðiÞ =rci þ sði1Þ Þaði1Þ ; for plastic softening zone Dði1Þ ¼ res res a rci ðmres ; for plastic residual zone b rrðiÞ =rci þ s Þ In the plastic residual zone, by incorporating Eq. (8b) into Eq. (11), one obtains:

ð28Þ

ði1Þ

where j is the number of the annulus immediately outside the plastic softening-residual boundary. In the calculation process, it is found that Eq. (25) can be derived as:

ehðiÞ ¼

uðiÞ C ði1Þ ðr ðiÞ =r ði1Þ  1Þ þ uði1Þ =r ði1Þ ¼ rðiÞ r ðiÞ =rði1Þ þ K wðiÞ ðr ðiÞ =rði1Þ  1Þ

ð29Þ

erðiÞ ¼

DuðiÞ r ðiÞ =rði1Þ  1 ¼ K wðiÞ ehðiÞ þ C ði1Þ  Dr ðiÞ 1  r ði1Þ =r ðiÞ

ð30Þ

Eqs. (23), (29) and (30) show that strain components in the plastic softening and residual zones are independent of Rp . Therefore, for a given analysis condition, the right side of Eq. (28) remains constant. Rp is linearly correlated to Rr .

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101

Fig. 13. Flow chart for the calculation process.

Table 2 Parameters for verification example. Parameters

Value

R0 (m)

3 0.25 15 5.7 30 25 2.0 0.6 0.004 0.002 1.698 1.191 0.01

l r0 (MPa) E (GPa) rcip (MPa) rcir (MPa) mpeak mres speak sres Kpw Krw

g⁄

hypothesised value of Rp is taken as an input boundary condition to make the tentative calculation possible. For VCSM, the calculation method can be summarised as follows: (i) Calculate rrðiÞ based on Eq. (16). When i ¼ 1, rrði1Þ ¼ rr2 . rr2 can be solved by Eq. (15) using the Newton–Raphson method. (ii) Solve rðiÞ and uðiÞ using Eqs. (23) and (25). When i ¼ 1, r ði1Þ ¼ Rp , and uði1Þ ¼ ð1þlÞðpE0 rr2 Þ  Rp (referring back to Eq. (14)). (iii) Calculate ehðiÞ and erðiÞ by Eqs. (29) and (30). Calculate rhðiÞ by the following equation:

rhðiÞ ¼ rrðiÞ þ rci ðmbði1Þ rrðiÞ =rci þ sði1Þ Þaði1Þ when i ¼ 1, 4.6. Verification of linear relations In order to validate the linear relations, 5 tunnel cases in Alejano et al. (2010) are further studied here. 5 cases are referred to as analysis conditions A1–E1 from hereon. The tunnels are embedded in different limestone rock masses (with different GSI values) at a depth 600 m below the ground surface. Table 1 lists the calculation details for these analysis conditions. The strength parameter a in H–B criterion is 0.5 for A1–E1. The support pressure pi is assumed to be 0. Due to the fact that the focus here is on the relations among Rp , Rr and R0 rather than on their numerical values, Rp is hypothesised to be 5 different values (4 m, 8 m, 12 m, 16 m and 20 m) here. Each

ð31Þ

rhði1Þ is equal to reh jr¼Rp ; ehði1Þ and erði1Þ can be

obtained by substituting ue jr¼Rp into Eq. (24). (iv) Calculate gðiÞ by accumulating DgðiÞ in Eq. (19) and gðiÞ by Eq. (10) (note that) gðiÞ ¼ g jr3 ¼rrðiÞ . Determine mbðiÞ , sðiÞ and aðiÞ based on Eq. (5). When i ¼ 1, gði1Þ ¼ 0, gði1Þ ¼ g jr3 ¼rr2 , mbði1Þ ¼ mpeak , sði1Þ ¼ speak and aði1Þ ¼ apeak . b (v) Judge whether or not gðiÞ P gðiÞ . If yes, stop the calculation loop and record the current value of rðiÞ (i.e.) Rr ; otherwise, set, i ¼ i þ 1 and repeat steps (i)–(v). (vi) In the plastic residual zone, calculate rðiÞ using Eq. (23) under res

0 res a the condition that Hðr0rðiÞ ; gði1Þ Þ ¼ rci ðmres Þ . b rrðiÞ =rci þ s

When rrðiÞ decreases to pi, r ðiÞ is equal to the calculated R0 corresponding to a hypothesised value of Rp .

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Fig. 14. Comparison between the calculated results and literature data (a) Ground response curve; (b) evolution of plastic softening and plastic residual radii and (c) distribution of the radial and tangential stresses.

Fig. 11a and b plot the variations of Rr and R0 versus Rp for analysis conditions A1–E1. It is shown that each curve shows a linear 2

trend (R ¼ 1). Moreover, one interesting observation is that for a given analysis condition, rr1 maintains a constant regardless of the hypothesised value of Rp (as shown in Fig. 12). This means Rr (and thus) Rp is linearly correlated to R0 since the ratio of Rr to R0 (as demonstrated in Eq. (27)) depends only on rr1 – a constant value. According to Eq. (16), the constant rr1 can also validate the fact that the g is not affected by Rp . These linear relations are fairly useful in solving aforementioned equations. To be specific, the constant rr1 can be calculated firstly. Then, Rr can be determined by Eq. (27). The true value of Rp can be gained based on the linear relationship between Rp and Rr by Eq. (28). 4.7. Calculation procedure To sum up, the calculation process can be divided into two stages. For the first stage (with the unidentified value of Rp ), the stress and strain components in the plastic softening and residual zones can be obtained. For the second stage (with the identified Rp ), the stress components in the elastic zone, and the deformation of the elastic and plastic zones can be finally solved out. The flow chart for the implementation of the proposed method to solve VCSM is illustrated in Fig. 13. 5. Verification example Lee and Pietruszczak (2008) and Wang et al. (2010) have analysed the same case by use of different methods. The rock mass was

Table 3 Values of g⁄aver for analysis conditions A1–E1. Analysis condition

A1

B1

C1

D1

E1

g⁄aver (103)

0.73

1.10

2.76

9.74

203.79

assumed to obey strain softening model and H–B failure criterion. Table 2 lists the values of relevant parameters. In this section, the case will be used to validate the applicability of the proposed method. Note that all the studies mentioned above (Lee and Pietruszczak, 2008; Wang et al., 2010) adopted CCSM to analyse the case, hence for the sake of comparability, the critical plastic softening parameter g⁄ in the proposed method is set to be constant for the whole plastic zone, i.e. the proposed VCSM is regressed to CCSM. The calculation strictly follows the procedure proposed above. Fig. 14a and b shows the ground reaction curve (GRC) as well as the evolution of Rp and Rr calculated from the proposed method and other two numerical methods in Lee and Pietruszczak (2008) and Wang et al. (2010). u0 is the radial displacement at the opening surface; G is the shear modulus of rock mass. It is observed that the results by the regressed version of the proposed method are highly consistent with those by the brittle plastic method (Wang et al., 2010) and finite difference method (Lee and Pietruszczak, 2008). This means the accuracy of the proposed calculation procedure is acceptable. The distributions of tangential stress rh and radial stress rr calculated by the regressed version of the proposed method and Lee and Pietruszczak’s method (Lee and Pietruszczak, 2008) are plotted in Fig. 14c. It can be seen that the curves by the

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103

Fig. 15. Variations of g and g⁄ in the plastic softening zone: (a) A1; (b) B1; (c) C1; (d) D1 and (e) E1.

two different methods are almost identical. Both rh and rr in the plastic softening zone rise more steeply than those in the plastic residual zone. This can be ascribed to the different variation trends of the strength parameters in the two zones.

6. Discussion In this section, both the VCSM and CCSM are used to carry out some parametric studies, in an effort to identify the necessity to consider the variable confining stress in the plastic zone. g⁄ in CCSM is calculated according to the method given by Alonso et al. (2008): first, solve rr2 by using Eq. (15); second, solve g⁄aver by substituting rr2 =2 into Eq. (10). Table 3 lists the values of g⁄aver for analysis conditions A1–E1 presented in Table 1. By comparing the results for the 5 analysis conditions calculated by VCSM and CCSM, the evolutions of g⁄, g, mb , rr and rh in the plastic zone, and the influence of pi on u0 and, Rp are discussed below.

6.1. Evolutions of g* and g Fig. 15a–e plot the variation of g and g⁄ versus rr . It should be noted in Fig. 15 that the red and green dashed lines represent the values of rr1 for VCSM and CCSM, and the red continuous lines represent values of rr2 for VCSM and CCSM. The results for VCSM illustrate that, as rr reduces from rr2 to rr1 , g ascends from 0 while g⁄ falls from the initial value, and both of them approach to the same value at the plastic softening-residual boundary. The value of g⁄ becomes larger and drops more sharply as the rock mass quality becomes worse. The reason is: rr2 is larger for lower GSI, and thus, the poorer quality of rock mass leads to a higher confining stress. On the basis of Eq. (10), g⁄ and its decreasing rate become greater with a larger confining stress. Especially, for GSIp = 40 and 25, g⁄ in VCSM falls so rapidly that, in the inner part of the plastic softening zone, it is even lower than the constant g⁄ in CCSM. Moreover, it is found that as g⁄ in VCSM is lower than that in CCSM, g becomes slightly greater in VCSM than in CCSM. In fact, a greater g⁄ gives

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Fig. 16. Variation of mb in the plastic softening zone versus g: (a) A1; (b) B1; (c) C1; (d) D1 and (e) E1.

rise to a larger strength parameter mb in a given annulus, so that accumulated deviatoric strain becomes smaller. Hence, g in VCSM will increase not so rapidly as that in CCSM. In addition, it should be mentioned that rr1 is heavily affected by g⁄ and its variation trend. This is because, according to Fig. 15a–e, the more sharply g⁄ drops, the faster g develops, the sooner the plastic softening-residual boundary will be reached,

which leads to a higher value of rr1 . Therefore, for analysis condition D1 (GSIp = 40), rr1 is higher in VCSM than in CCSM. For analysis condition E1 (GSIp = 25), no plastic residual zone is formed since g increases slowly. This is different from the other 3 analysis conditions. On the whole, it can be concluded that both the variation of g and rr1 are affected by the magnitude and variation rate of g⁄.

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Fig. 17. Distributions of rr and rh in the plastic zone: (a) A1; (b) B1; (c) C1; (d) D1 and (e) E1.

The comparison between the results (rr1 , g⁄ and g) calculated by CCSM and VCSM for the lower GSIp conditions (GSIp = 25, 40) is different from that for the other 3 conditions. 6.2. Evolution of mb in the plastic softening zone Fig. 16a–e plot the variation of mb versus g for the 5 analysis conditions. It should be noted in Fig. 16 that the red and green dashed lines represent the value of mb at the plastic softening-residual boundary. Since mb , s and a are calculated by the same equation (Eq. (6)), the variation trend for mb is able to represent that for s and a. In accordance with Eq. (6), the evolution of mb in CCSM declines linearly with g in the plastic softening zone. This is also reflected in Fig. 16a–e. As for VCSM, when GSIp changes from 75 to 60, the curves approximately show linear trends. In contrast, as GSIp further reduces from 60 to 25, the curves become nonlinear and the decreasing rate of mb gradually grows with the

increase in g. Hence, for GSIp = 40, mb calculated by CCSM surpasses that by VCSM when g increases to a certain value. This ‘‘cross-over’’ phenomenon is consistent with the trend of g⁄ depicted in Fig. 15d. Especially, for GSIp = 25, when g is within a certain value, mb of VCSM is basically equal to that of CCSM, whereas beyond this value, mb in CCSM becomes substantially larger than CCSM as g increases.

6.3. Distributions of

rr and rh

Fig. 17a–e demonstrates the variations of rr and rh calculated by CCSM and VCSM for the 5 analysis conditions. It should be noted in Fig. 17 that the red and green dashed lines represent the values of Rr for VCSM and CCSM, the red and green lines represent the values of Rp for VCSM and CCSM. It is easy to see that rr (i.e. the confining stress) increases but rh decreases with the decrease in GSIp.

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Fig. 18. Influence of pi on u0 and Rp for VCSM and CCSM: (a) u0 versus pi with A1–E1; (b) Rp versus pi with A1–E1; (c) u0 versus pi with A1–D1 and (d) Rp versus pi with A1–D1.

To be more specific, a lower value of GSIp leads to a higher value of rr in the elastic zone (referring to Eq. (15)). As for rh , it is expressed in Eq. (3) that rh is calculated by rr and the strength parameter x. Actually, it is recognised that x is continuously decreasing with the decrease in GSIp. This is the main reason why rh decreases. Hence, it is concluded that rh is more easily affected by the strength parameter x.Fig. 17a–e also indicate that for analysis conditions A1–C1, the calculated Rr and Rp are greater in CCSM than in VCSM, and the calculated rh in the plastic softening zone is evidently lower in CCSM than in VCSM. The comparison shows inverse trends for analysis conditions D1–E1. According to Eq. (27), as rr1 in CCSM is evidently lower than that in VCSM, Rr for analysis condition D1 is smaller in CCSM than in VCSM, contrary to that in analysis conditions A1–C1. For rh , since rr of CCSM and VCSM are basically the same, the variation of rh tends to be consistent with the variation of x (referring back to Eq. (3)). Therefore, similar to the variations of x for analysis conditions A1–E1, rh in the plastic softening zone is lower in VCSM than in CCSM when GSIp = 40 and 25, opposite to the other 3 analysis conditions. To conclude, values of rr calculated by VCSM and CCSM are approximately the same in the plastic zone. In contrast, difference in the values of rh by VCSM and CCSM is evident. In the plastic zone, rr (i.e. the confining stress) decreases but rh increases with the increase in GSIp. The former is mainly influenced by the strength parameters in the elastic zone, while the latter by those in the plastic zone. 6.4. Influence of pi on u0 and Rp For analysis conditions A1–E1, u0 and Rp under different values of pi , are shown in Fig. 18a and b. Fig. 18c and d are the

Table 4 Values of u0, Rp and Rr calculated by VCSM and CCSM under different pi. GSI

pi (MPa)

u0 (m)

Rp (m)

Rr (m)

CCSM

VCSM

CCSM

VCSM

CCSM

VCSM

75

0.0 0.2 0.4 0.6 0.8 1.0

0.21 0.18 0.17 0.16 0.15 0.15

0.21 0.19 0.17 0.16 0.15 0.14

3.56 3.38 3.26 3.17 3.10 3.04

3.57 3.39 3.27 3.18 3.11 3.04

3.52 3.35 3.23 3.14 3.07 3.00

3.50 3.33 3.21 3.12 3.05 3.00

60

0.0 0.2 0.4 0.6 0.8 1.0

0.60 0.53 0.49 0.46 0.43 0.41

0.59 0.52 0.48 0.45 0.42 0.40

4.17 3.91 3.76 3.65 3.55 3.46

4.21 3.96 3.80 3.68 3.59 3.50

3.99 3.75 3.60 3.49 3.40 3.31

4.14 3.89 3.74 3.62 3.52 3.44

50

0.0 0.2 0.4 0.6 0.8 1.0

1.17 1.01 0.92 8.56 0.81 0.76

1.09 0.94 0.88 0.81 0.79 0.75

4.59 4.26 4.07 3.93 3.82 3.72

4.56 4.23 4.04 3.90 3.79 3.69

4.12 3.82 3.65 3.53 3.42 3.33

4.09 3.79 3.63 3.50 3.40 3.31

40

0.0 0.2 0.4 0.6 0.8 1.0

2.13 1.83 1.66 1.54 1.45 1.37

2.28 1.86 1.70 1.58 1.48 1.40

4.89 4.51 4.30 4.14 4.02 3.92

4.95 4.56 4.34 4.19 4.06 3.95

3.44 3.17 3.02 – – –

4.04 3.73 3.55 3.42 3.32 3.23

25

0.0 0.2 0.4 0.6 0.8 1.0

6.32 5.29 4.78 4.42 4.13 3.89

7.23 5.78 5.09 4.61 4.23 3.94

6.53 5.77 5.39 5.11 4.89 4.71

6.056 5.50 5.21 5.00 4.83 4.68

– – – – – –

3.97 3.51 3.28 3.11 3 3

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Fig. 19. Influence process of GSIp in VCSM.

corresponding enlarged views for A1–E1. The relevant parameters and calculated results are listed in Table 4. For each condition, pi varies from 0 to 1 MPa. As shown in Fig. 18a, the inconsistency between the curves of u0 versus pi in VCSM and CCSM becomes more evident with the decrease in GSI and pi . When GSIp = 40 and 25 with no support pressure, u0 in VCSM are 6.83% and 14.40% higher than that in CCSM, respectively. By contrast, when GSIp = 50, u0 in VCSM becomes slightly lower than that in CCSM. As GSIp grows (GSIp = 60, 75), the difference between the two models becomes marginal. The differences of Rp by VCSM and CCSM show the same tends with that of u0 for analysis conditions A1–E1, whereas the trends of Rp are more evident. The internal parameters (such as rh , rr , g⁄, g⁄, m) can be used to explain the different distributions of u0 in VCSM and CCSM. Essentially, g⁄ is the most important parameter governing the strain-softening behaviour. When GSIp is 25 or 40, the confining stress is large enough, resulting in a fairly high value of g⁄. Thus, the initial value and decreasing rate of g⁄ in VCSM are pretty great, as a result, g⁄ in CCSM will exceed that in VCSM at a certain radius during the plastic softening stage. This will lead to relatively smaller values of Rr in CCSM. Additionally, it can be seen from Table 4 that under a certain condition, the contrast of Rp calculated by CCSM and VCSM is in line with that of Rr . Therefore, a higher g⁄ can give rise to higher Rp and Rr together. On the other hand, when GSIp = 25 or 40, strength parameters solved by VCSM falls more dramatically after g⁄ of CCSM surpasses that of VCSM. Hence, lower strength parameters with lower g⁄ will be obtained. Due to the fact that VCSM causes a lower g⁄, the strength parameters of VCSM are smaller, and its Rp and Rr of VCSM are greater. This is the reason why CCSM underestimates u0 compared to VCSM when GSIp is 25 or 40. In fact, Lee and Pietruszczak (2008) and Wang et al. (2011) have demonstrated their view – the higher g⁄, the lower u0. This can confirm the analysis here, though g⁄ is regarded as unchanged in Lee and Pietruszczak (2008) and Wang et al. (2011). In summary, compared to VCSM, CCSM underestimates the displacement u0 for the poorer rock mass investigated (GSIp = 25 and 40), whereas it slightly overestimates u0 for better rock masses (GSIp = 50). It is the initial value of g⁄ that affects the strength parameters, softening parameter, stress distribution, radii of the plastic softening and residual zones, and radial displacement. The influence process can be described in Fig. 19. 7. Conclusions Based on the relation between the confining stress r3 and the critical plastic softening parameter g⁄ proposed by Alonso et al.

107

(2008), the variable confining stress model (referred to as VCSM) was first introduced to account for the influence of variable confining stress on the elasto-plastic behaviours. A new numerical procedure was then presented for VCSM. The verification example indicated that the proposed procedure is capable of providing reasonable estimation. In the end, a series of parametric studies was carried out by use of CCSM and VCSM. The main conclusions include: The critical plastic softening parameter g⁄ is a fundamental parameter governing the strain-softening behaviour of rock mass. The effect of g⁄ on the stability of a circular opening in strain-softening rock mass includes two different aspects. For one thing, g⁄ governs the strength parameters. For another, g⁄ affects Rp and Rr by determining the time reaching the plastic softening-residual boundary. The difference in the radial displacement between VCSM and CCSM is more obvious for lower GSIp, and it becomes marginal as GSI and support pressure are relatively great. Compared to VCSM, CCSM may slightly overestimate the radial displacement for rock mass with comparatively better quality (such as GSIp = 50 in this study), but underestimate the radial displacement for rock mass with poor quality (such as GSIp = 40, or 25 in this study). Therefore, attention should be paid to this issue in the design practice of rock and mining engineering. In the plastic zone, radial stress rr (i.e.) r3 decreases but tangential stress rh increases with the increase in GSIp. Values of rr calculated by VCSM and CCSM are almost the same in the plastic zone. Nonetheless, as rh is directly related to the strength parameters in the plastic zone, the difference between the values of rh calculated by VCSM and CCSM is more significant. Acknowledgement The authors acknowledge the financial support provided by the Program for New Century Excellent Talents, Ministry of Education, PR China (No. NCET-06-0649). References Alejano, L.R., Alonso, E., 2005. Considerations of the dilatancy angle in rocks and rock masses. Int. J. Rock Mech. Min. Sci. 42, 481–507. Alejano, L.R., Alonso, E., Rodriguez-Dono, A., Fernandez-Manin, G., 2010. Application of the convergence-confinement method to tunnels in rock masses exhibiting Hoek–Brown strain-softening behaviour. Int. J. Rock Mech. Min. Sci. 47, 150– 160. Alejano, L.R., Rodriguez-Dono, A., Veiga, M., 2012. Plastic radii and longitudinal deformation profiles of tunnels excavated in strain-softening rock masses. Tunn. Undergr. Space Technol. 30, 169–182. Alonso, E., Alejano, L.R., Varas, F., Fdez-Manin, G., Carranza-Torres, C., 2003. Ground response curves for rock masses exhibiting strain-softening behaviour. Int. J. Numer. Anal. Meth. Geomech. 27, 1153–1185. Alonso, E., Alejano, L.R., Fdez-Manin, G., Garcia-Bastante, F., 2008. Influence of postpeak properties in the application of the convergence-confinement for designing underground excavations. In: Proceedings of the 5th International Conference and Exhibition on Massive Mining Technology, Lulea, Sweden, pp. 793–802. Brown, E.T., Bray, J.W., Ladanyi, B., Hoek, E., 1983. Ground response curves for rock tunnels. J. Eng. Mech. ASCE 109, 15–39. Cai, M., Kaiser, P.K., Uno, H., Tasaka, Y., Minami, M., 2004. Estimation of rock mass deformation modulus and strength of jointed hard rock masses using the GSI system. Int. J. Rock Mech. Min. Sci. 41, 3–19. Cai, M., Kaiser, P.K., Uno, H., Tasaka, Y., Minami, M., 2007. Determination of residual strength parameters of jointed rock masses using the GSI system. Int. J. Rock Mech. Min. Sci. 44, 247–265. Carranza-Torres, C., 1998. Self Similarity Analysis of the Elastoplastic Response of Underground Openings in Rock and Effects of Practical Variables. Ph.D Thesis. University of Minnesota. Carranza-Torres, C., Fairhurst, C., 1999. The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion. Int. J. Rock Mech. Min. Sci. 36, 777–809. Carranza-Torres, C., Fairhurst, C., 2000. Application of convergence-confinement method of tunnel design to rock masses that satisfy the Hoek–Brown failure. Tunn. Undergr. Space Technol. 15 (2), 187–213.

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