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International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms

Elasto-plastic analysis of a circular opening in strain-softening rock mass Qiang Zhang a,b,n, Bin-Song Jiang a, Shui-lin Wang b, Xiu-run Ge b, Hou-quan Zhang a a State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221008, PR China b State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 September 2010 Received in revised form 15 September 2011 Accepted 13 November 2011 Available online 17 December 2011

Geo-materials generally show strain-softening behavior after peak-load. To avoid the difﬁculty of describing variable material properties during the post-failure process in a strain-softening rock mass, a multi-step brittle–plastic model (MBPM) was proposed on the assumption that rock mass properties were uniform in a very small region. Then the post-peak rock mass can be divided into a number of annular regions and the rock mass in each region can be considered to conform to classical plasticity theory. For the strain-softening behavior, material properties of rock mass such as strength and deformation parameters were assumed to follow a piecewise linear function of plastic shear plastic strain. Then the equations for solving the annulus’ radius were given on the basis of the material properties and its corresponding plastic shear plastic strain in each annulus, and the radii were calculated in a successive manner by combining the secant and Newton–Raphson method from the outmost one whose initial radius is assumed. Several sets of examples are analyzed to validate the new approach. For elasto-brittle–plastic behavior, the results show high consistency with closed-form solutions. For the strain-softening behavior, the new approach also shows high consistency with the numerical results. Finally, the inﬂuence of shear modulus on the rock mass deformation was analyzed and the results show that it can affect the range of residual and softening radius lightly but inﬂuence the rock mass deformation heavily, which usually occurs in the geo-engineering, especially in the deep underground engineering. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Strain-softening Shear modulus deterioration Yield criterion Elasto-plastic analysis Analytical solution Circular tunnel

1. Introduction The stress and deformation states of surrounding rock in underground engineering are very important for excavation optimization design and stability evaluation. Although the elasto-plastic problem of a circular tunnel is simple for isotropic and strain-softening rock masses under hydrostatic stress, the ground reaction curve (GRC) of surrounding rock is also very important in geo-engineering and it can lay theoretical foundation for the (quasi-) circular tunnels such as TBM tunnel and circular diversion tunnel. In the past, many researchers [1–17] have done research on circular tunnel problems in an elasto-plastic, elasto-brittle–plastic and strain-softening way. Generally, the Mohr–Coulomb (M–C) criteria, Hoek–Brown (H–B) criteria and generalized Hoek–Brown (GHB) criteria are employed as associated and non-associated potential ﬂow laws. On this basis, the surrounding rock stress and

n Corresponding author at: State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221008, PR China. Tel.: þ86 15172477586. E-mail address: [email protected] (Q. Zhang).

1365-1609/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2011.11.011

deformation distribution is analyzed by ideal elasto-plastic and elasto-brittle–plastic mechanical models. Park and Kim [1] analyzed the surrounding rock stress and deformation law based on the different elastic strain distributed law of the plastic rock mass. Sharan [2] obtained the closed-form expressions of stress and deformation in elasto-brittle–plastic Hoek–Brown rock mass for circular tunnels. After the GHB criteria is proposed [3], only a few attempts have been made thus far for the mathematical difﬁculties in deriving neat with a a0.5 (a is an parameter of H–B criteria based on the Geological Strength Index). Due to the difﬁculty with generalized coefﬁcient a, Carranza-Torres [4] presented a rigorous elasto-plastic solution for the axi-symmetrical problem of excavating in GHB materials by re-writing the GHB criterion in terms of transformed stress quantities. Meanwhile, Sharan [5] obtained the approximate stress and deformation analytical solutions with GHB criteria on the assumption that the elastic strains are the same in the solution process of the H–B criteria and generalized one in plastic region. The given calculation example shows that the error between the approximate method and the closed-form solution is allowable for engineering. Jiang et al. [6] obtained the stress and deformation solution of circular tunnel in elasto-plastic–brittle rock mass on account that the deep underground rock mass possesses a higher plastic

Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 50 (2012) 38–46

deformation ability. Park et al. [7] analyzed the surrounding rock mass deformation and stress distribution law with different softening indexes and dilatant characters in the plastic region. But it is known that the surrounding rock conforms to the generalized Hooke’s law, so other solutions except case 3 in that paper are not correct. Meanwhile, the numerical method needs to be employed in the solution for the strain-softening process. Zheng et al. [8] proved that convergence problem exists in solving the strain-softening displacement by ﬁnite element method. So the post-failure rock mass was usually divided into ﬁnite annuluses and each annulus was considered as isotropic and uniform material. Brown et al. [9] obtained the analytical solution of circular tunnel in strain-softening rock mass by assuming that the elastic strain in plastic region is uniform and equivalent with that at the elastoplastic interface. Yuan and Chen [10] assumed negative elastic modulus in the post-peak region, and given the analytical solutions of circular openings in strain-softening rock mass. However, the essential reason of strain-softening is the deterioration of rock mass strength’s deterioration, but not the elastic modulus. Wang et al. [11] obtained numerical solution by dividing the surrounding rock mass into numbers of annuluses, which obeys ideal brittle–plastic mechanical model. Lee and Pietruszczak [12] also obtained the numerical solution using different methods by dividing the surrounding rock mass into numbers of annuluses. Carranza-Torres [13] also obtained the solutions of circular tunnel in strain-softening rock mass with self similar analysis. Further researches also have been done by others [14–19]. From the above, it can be found that the former researches were mainly focused on the analytical solutions for the circular tunnel in ideal elasto-plastic and elasto-brittle–plastic rock mass, or the numerical solutions in strain-softening rock mass. But there rarely exists any discussion about the analytical analysis in the strain-softening process after peak-load. It is known that the geotechnical materials perform generalized strain-softening behavior, so it is not appropriate to analyze the surrounding rock with perfect elasto(-brittle)–plastic mechanical model, which is a special case of strain-softening model. The actual rock mass failure process is the deteriorative and losing process of rock mechanical parameters such as elastic modulus, cohesive strength and friction angle, etc. as shown in Fig. 1 [20]. The former studies have considered the strength deterioration, but the deformation deterioration has not attracted attention in the strain-softening process as far. So, here we pay attention on the analysis of stress distribution and radial displacements of a circular tunnel in a strain-softening rock mass, using analytical method considering both the strength deterioration and Young’s modulus deterioration.

Fig. 1. Typical cyclic loading deformation curve of a rock sample.

39

2. Problem description 2.1. Multi-step brittle–plastic mechanical model Based on the complete strain–stress curve of a rock sample, the post failure behavior of geo-materials can be divided into two types: ideal elasto-plastic and strain-softening/hardening. As we know, the elasto-plastic and elasto-brittle–plastic models are just two typical and special types of strain-softening behavior, and they can be easily implemented with both numerical and analytical methods. Because the material properties of rock mass closely related with the plastic deformation of rock mass in the strain-softening model are variable, the difﬁculty of solving the stress and displacements increases signiﬁcantly. In order to obtain the analytical solutions of circular tunnels in strain-softening rock mass, it is assumed that the surrounding rock is divided into residual region, softening region and elastic region, and the post-peak rock mass is further divided into ﬁnite concentric annuli (Multi-step Brittle–Plastic Model, MBPM), as shown in Fig. 2. 2.2. Basic theory and equations Fig. 2 shows a circular tunnel of radius R0 excavated in inﬁnite strain-softening rock mass. And an initial hydrostatic stress state of p is imposed over the domain. The internal support pressure is s0, which is lower than a critical value sk in order to form a plastic region around the tunnel. The post-peak rock mass is composed of k sufﬁciently small annuli, where the ith annulus is bounded by two circles of radii Ri 1 and Ri. In each annulus the material properties are isotropic, uniform and unchangeable, which satisfy the classical plasticity theory. The residual and softening region are denoted by annulus with radii of R1 and Rk, respectively. In the kth annulus whose stress is namely at the peak value, the material property of rock mass is the same as that of intact rock mass in the elastic region. The calculated mechanical model is shown in Fig. 3. Additionally, the plane strain condition is assumed in this paper. In the polar coordinate, radial and tangent stresses and strains of surrounding rock can be denoted as sri, syi, eri and eyi, respectively. The radial displacement is denoted as ui. The subscript i denotes the

Fig. 2. Analytical model of circular opening in strain softening rock mass.

40

Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 50 (2012) 38–46

Fig. 3. Mechanical state model of rock mass.

Fig. 4. Evolution of strength parameters in post-peak region.

ith annulus, so the post-failure region is composed of annuli of 1st to (k1)th, while the kth annuli represents the elastic region. Meanwhile, the superscript i denotes the corresponding variable value at the r¼Ri. The default value of i varies from 1 to k 1 when there is no interpretation. The stresses satisfy the equilibrium equation in each annulus (assuming the body force as zero):

where Gi and li are rock mass shear modulus and plastic strain, respectively, and m is the Poisson’s ratio of rock mass. When li ¼0, Eq. (7) is the elastic constitutive equation. The rock mass fails when rock mass reaches elastic stress limit, then the rock mass strength and deformation parameters are deteriorated with the plastic deformation, which causes the poststrength surface shrinking. Generally, it is assumed that those parameters are described by bilinear function of the plastic shear plastic strain, which can be denoted as

dsri sri syi þ ¼0 dr r

ði ¼ 1,2 ,kÞ

ð1Þ

When the stresses of rock mass in plastic region satisfy the Mohr– Coulomb criterion and Hoek–Brown criterion, the yield criterion can be, respectively, expressed as F MC ¼ syi N i sri Si i pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ HB Fi ¼ syi sri sci mi sri =sci þ si

ð2Þ

where sci is the unaxial strength of the ith annulus of rock mass, mi and si are strength parameters of the H–B rock mass in the ith annulus. And Ni and Si are material parameters of the M–C rock mass, which can be expressed as N i ¼ ð1 þ sin fi Þ=ð1sin fi Þ Si ¼ 2C i cos fi =ð1sin fi Þ

ð3Þ

where Ci and fi are cohesive strength and friction angle of M–C rock mass, respectively. In each annulus, the geometric equations can be expressed as i eri ¼ du dr ui eyi ¼ r

ð4Þ

In the elasto-plastic problem, the complex nonlinear deformation is caused by the plastic component, which is lied on the plastic potential. Meanwhile, the plastic potential is generally in accordance with the Mohr–Coulomb potential for geo-materials [21].

Fi ¼ syi ai sri

ð8Þ

In this paper, bi is predetermined as a priori variable and the corresponding variable of material parameters can be determined in each annulus. Therefore, all of the physical parameters, except for stress, strain and deformation, are known as constants in each annulus. The material parameters evolution law of rock mass can be obtained by a series of unaxial and triaxial loading–unloading tests and the practical estimate method can also be considered [22]. In this paper, the material parameters of rock mass are considered to follow a multi-linear function of plastic shear plastic strain, as shown in Fig. 4. Then the surrounding rock mass physical parameters can be expressed as wi ¼ wk

wk w1

bn

bi

ð9Þ

where wi represents one of the strength and deformation parameters, such as Ci, ji, ci, mi, si, sci and xi for rock mass. The parameter xi ¼Gi/ Gk is the softening factor of shear modulus for rock mass in the ith annulus. The stress boundary conditions and contact conditions are shown as follows:

sr ¼ s0 r ¼ Ri , srði þ 1Þ ¼ sri ,ui þ 1 ¼ ui r-1, sr ¼ p r ¼ R0 ,

ð10Þ

3. Stress and deformation solutions ð5Þ 3.1. The stress and deformation of M–C rock mass

where ai is material deformation parameter, namely

ai ¼ ð1 þsin ci Þ=ð1sin ci Þ

bi ¼ epyi epri

ð6Þ

where ci is rock mass dilatancy angle. Based on the plastic potential Eq. (5), using non-associated ﬂow rule, the total strain caused by excavation can be expressed as eri ¼ 2G1 i ð1mÞðsri pÞmðsyi pÞ li ai ð7Þ eyi ¼ 2G1 i ð1mÞðsyi pÞmðsri pÞ þ li

In the post-failure region (Ri 1 or oRi, i¼1,2yk), using the equilibrium equation (1), the ﬁrst equation of yield criterion Eq. (2) and the ﬁrst equation of boundary condition Eq. (10), the stress can expressed as Ni 1 r sri ¼ sci1 Ri1 Di ð11Þ Ni 1 r syi ¼ N i sci1 Ri1 Di

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where sci1 ¼ si1 þ Di , Di ¼ C i cot fi , and si 1 is the radial stress at r ¼Ri 1, namely, si1 ¼ sci2 ðRi1 =Ri2 ÞNi1 1 Di1 . Submitting the geometric equation (4) and stress expression equation (11) into constitutive equation (7), the plastic deformation li can be obtained T i Ri 1 þ ai sci1 r Ni 1 li ¼ þ ð1Ni Þð1 þ Zi Þ ð12Þ Ri1 2Gi r 2Gi where Zi ¼(1 m)(1 þNiai)/(Ni þ ai) m, and Ti is the integration constant. Substituting Eq. (12) into constitutive equation (7), the strain and displacement can be obtained as

Ni 1 ai þ 1 r eri ¼ 2G1 i N i Zi sci1 Ri1 þT i ai Rri ð12mÞpi

Ni 1 ai þ 1 ð13Þ r eyi ¼ 2G1 i Zi sci1 Ri1 T i Rri ð12mÞpi ui ¼ r eyi where pi ¼p þDi. In the elastic region (Rk rr oN), the stress and displacement can be easily obtained, namely

2 2 srðk þ 1Þ ¼ p 1 Rrk þ sk Rrk ð14Þ 2

2 syðk þ 1Þ ¼ p 1 þ Rrk sk Rrk

erðk þ 1Þ ¼

1 2Gk

eyðk þ 1Þ ¼

1 2Gk

ðsk pÞ k

Rk r

ak C k R1 ¼ ð12mxk Þðpsk Þdk Bk 2dk Ak ln k ai C i R1 ¼ i

Rk 2d A þ k k Rk1 1 þ ak

i xi h 1ai þ 1 C R þð12mÞðsi pÞ þdi þ 1 Bi þ 1 2di þ 1 Ai þ 1 =ð1 þ ai þ 1 Þ xi þ 1 i þ 1 i ð12mÞðsi pÞ2di Ai ln

Ri þ di Bi þ 2di Ai =ð1 þ ai Þ Ri1

ð20Þ

4. Rock mass re-rupturing condition It is known that the rock mass material strength and deformation parameters are closely related with the plastic shear plastic strain b, inﬂuencing the rock mass deformation heavily. Based on the plastic potential theory, the plastic strain components can be expressed as

epri ¼ p

eyi ¼

@Fi @sri

¼ li ai

@Fi @syi

¼ li

bi ¼ ð1 þ ai Þli

2

ð15Þ

Rk r

uðk þ 1Þ ¼ r eyðk þ 1Þ Using the continuous conditions at the interface of adjacent softening annulus, namely, the second equation of Eq. (10), the integration constant Ti can be obtained as Nk 1 k ð12m þ xk Þpk T k ¼ ðZk þ xk Þsck1 RRk1 1 þ ai þ 1 Ni 1 T i ¼ T i þ 1 xixþi 1 RiRþi 1 þ sci1 Zi xixþi 1 Zi þ 1 RiRþi 1 xi x þ ð12mÞ pi þ 1 pi i Zi þ 1 ðDi þ 1 Di Þ ð16Þ

xi þ 1

where d1i ¼(12m)Ai, d2i ¼2diAi þ(1 2m)Bi, d3i ¼(1 2m)(si 1 p)þ diBi 2diAi/(1þ ai). In the elastic region (Rk rr oN), the stress and displacement are the same as that of M–C rock mass, namely, Eqs. (14) and (15). In the same way, the integral constant Ci can be obtained using the second equation of Eq. (10), as shown in Eq. (20)

ð21Þ

Substituting Eq. (21) into Eq. (8), the plastic shear plastic strain can be obtained, namely

2

ðps Þ

41

xi þ 1

3.2. The stress and deformation of H–B rock mass In the post-failure region (Ri 1 oroRi, i¼1,2yk), with the same analysis method of using Eq. (1), the second equation of Eq. (2) and the ﬁrst expression of Eq. (10), the stress can be expressed as r r sri ¼ Ai ln2 Ri1 þ Bi ln Ri1 þ si1 r r syi ¼ Ai ln2 Ri1 þ ð2Ai þ Bi Þln Ri1 þ Bi þ si1

ð22Þ

The rock mass properties is uniform in each annulus, but once the rock mass deformation satisﬁes a certain condition, the rock mass reruptures with the deterioration of material properties. Then the mechanical state of rock mass transits from the ith annulus to the (i 1)th annulus. As shown in Fig. 4, the rock mass re-rupturing condition can be described by the points’ plastic shear plastic strain of A1–A4, corresponding to the cases 1–4 in the sequel.

4.1. M–C rock mass re-rupturing condition Case 1. When the maximum plastic shear plastic strain of rock mass in the (iþ1)th annulus, namely, the plastic shear plastic strain at the interface with the second internal annulus reaches the predetermined value of ith annulus, the rock re-ruptures. Using Eqs. (8) and (22), we have R i 1 þ ai ð1Ni ÞðZi þ 1Þsci1 T i ð1 þ ai Þ 2Gi bi1 ¼ 0 ði ¼ 2,3 ,kÞ Ri1 ð23Þ

ð17Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 where Ai ¼misci/4, Bi ¼ mi sci si1 þ si s2ci , and si1 ¼ Ai1 ln ðRi1 = Ri2 Þ þ Bi1 lnðRi1 =Ri2 Þ þ si2 . Using the geometric equation (4), constitutive equation (7) and stress expression (11), the plastic deformation li can be denoted as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1mÞ di C mi sci sri þ si s2ci A þ i r ð1 þ ai Þ li ¼ ð18Þ Gi ð1 þ ai Þ Gi ð1þ ai Þ i 2Gi where di ¼(1 m)(ai 1)/(1þ ai), and Ci is the integration constant. In the same way, substituting Eq. (18) into Eq. (7), the strain and displacement can be denoted as

1 r 2 r eri ¼ d1i ln þ ð2d1i þ d2i Þln þ d2i þ d3i C i ai r 1ai 2Gi Ri1 Ri1

1 r 2 r eyi ¼ d1i ln þd2i ln þd3i þC i r 1ai ui ¼ r eyi ð19Þ 2Gi Ri1 Ri1

At the interface between elastic region and softening region, the elastic stress satisﬁes the peak yield condition. Then we get sy(k þ 1) ¼ syk at r ¼Rk. Using Eqs. (11) and (14), we can obtain that ðN þ 1Þsck1 ðRk =Rk1 ÞNk 1 2pk ¼ 0

ð24Þ

Case 2. It is assumed that the material properties of rock mass in each annulus are represented by the starting point r ¼Ri at each step. Fig. 4 shows that the properties of (iþ1)th annulus are represented by point A2. So post-peak rock mass re-rupturing condition can be expressed as ð1Ni ÞðZi þ 1Þðsi þDi ÞT i ð1 þ ai Þ2Gi bi ¼ 0

ð25Þ

Case 3. In the same way, when each annulus’ material properties are represented by its middle point r ¼(Ri þ Ri 1)/2, such as A3 in the ith annulus, the post-peak rock mass re-rupturing condition

42

Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 50 (2012) 38–46

can be expressed as 1 þ ai R þ R Ni 1 2Ri sci1 ð1Ni ÞðZi þ 1Þ i1 i T i ð1 þ ai Þ 2Gi bi ¼ 0 2Ri1 Ri1 þ Ri ð26Þ Case 4. When each annulus’ material properties are represented by its end point r ¼Ri 1, such as A4 in the ith annulus, the postpeak rock mass re-rupturing condition can be expressed as Ri 1 þ ai sci1 ð1Ni ÞðZi þ 1ÞT i ð1 þ ai Þ 2Gi bi ¼ 0 ð27Þ Ri1

4.2. H–B rock mass re-rupturing condition

As for the H–B rock mass, the four post-rock mass re-rupturing conditions can be obtained in the same way. Case 1: when the material properties are represented by the maximum deviator plastic strain of the former annulus, the rerupturing condition of H–B rock mass can be expressed as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C i ð1 þ ai ÞRi1 ð1 þ ai Þ 2½ð1mÞ mi sci si1 þsi s2ci þ di Ai 2Gi bi1 ¼ 0

Fig. 5. Post-failure rock mass unloading curves.

5.1. Threshold value of x1 governed by M–C criteria

Additionally, at the interface between elastic region and softening region, the elastic stress should satisfy the H–B yield criterion, thus we have " sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ # Rk 1 mk 2 mk p m 2 Rk k1 Ak ln þ Bk ln þ s pþ sck þ þ sk k ¼ 0 2 Rk1 Rk1 4 sck 8

At each interface of adjacent annulus, the increment of deviator plastic shear strain of M–C rock mass can be expressed as 1 x Dbi ¼ ðai þ 1 ai ÞT i ð1 þ ai þ 1 Þ Zi i Zi þ 1 ðsi þ Di Þ 2Gi xi þ 1

xi xi pi þ 1 pi Zi þ 1 ðDi þ 1 Di Þ þ ð12mÞ xi þ 1 xi þ 1 x þ ð1Ni Þð1 þ Zi Þðsi þDi Þ i ð1Ni Þð1 þ Zi þ 1 Þðsi þ Di þ 1 Þ

ð29Þ

ð34Þ

When the material properties of rock mass in the ith annulus are represented by the starting point, middle point and end point, such as A2, A3 and A4 of the ith annulus, the re-rupturing conditions of cases 2–4 can be, respectively, shown as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C i ð1 þ ai ÞRi ð1 þ ai Þ 2½ð1mÞ mi sci si þ si s2ci þdi Ai 2Gi bi ¼ 0 ð30Þ

It is known that Dbk should also be positive when Rk ¼Rk 1, however, it is unknown whether Rk ¼Rk 1. It is known that the accumulate deviator plastic strain in the kth annulus is nonnegative when Rk aRk 1, and bk ¼ Dbk when Rk ¼ Rk 1. So the Drucker assumption can be satisﬁed by the positive deﬁniteness of total deviator plastic shear strain (bk Z0) in the kth annulus, which can be expressed as follows: T k Rk 1 þ ak sck1 bk ¼ ð1 þ ak Þ þ ð1Nk ÞðZk þ 1Þ ð35Þ 2Gk Rk1 2Gk

ði ¼ 2,3 ,kÞ

C i ð1 þ ai Þ

ð28Þ

Ri1 þ Ri ð1 þ ai Þ R þ Ri 2ð1mÞ 2Ai ln i1 þ Bi 2di Ai 2Gi bi ¼ 0 2 2Ri1

ð31Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C i ð1 þ ai ÞRi1 ð1 þ ai Þ 2½ð1mÞ mi sci si1 þ si s2ci þ di Ai 2Gi bi ¼ 0

xi þ 1

ð32Þ 5.2. Threshold value of x1 governed by H–B criteria

5. Determination of threshold value n1 Generally, the cracked rock Young’s modulus deteriorates, but the rock mass deterioration degree cannot be inﬁnite, and it should satisfy a certain condition. Fig. 5 shows unloading curves of the postfailure rock mass at the interface between (iþ1)th and ith annulus. At point A1, it unloads along the line L1, the corresponding plastic p (shear) strain is epi1 ðbi1 Þ. Meanwhile, the unloading process will develop along with the lines of L2 when shear modulus deterioration factor xi is about 1 at point A2. Then, the plastic (shear) strain is ep1i ðbp1i Þ. However, when xi is signiﬁcantly smaller than xi þ 1, it would unload along with the line of L3, so the plastic (shear) strain should p be ep2i ðb2i Þ. In this condition, Fig. 5 shows that the plastic (shear) strain of the ith annulus is smaller than that of the (iþ 1)th annulus, p p namely, epi1 4 ep2i ðbi1 4 b2i Þ, which is not satisﬁed with the Drucker assumption. So the deviator plastic strain should be nondecreased during the post-failure process. Then we have

Dbi 40

ð33Þ

where Dbi is the increment of deviator plastic strain in the ith annulus.

In the same way, the Drucker assumption of H–B rock mass can be ascertained as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 x Dbi ¼ i ð1mÞð1þ ai þ 1 Þ mi sci si þsi s2ci 2Gi þ 1 xi þ 1 xi ð12mÞð1 þ ai Þðsi pÞ þ 1 xi þ 1

a a ð1 þ ai Þðai þ 1 1Þ þ 2di þ 1 Ai þ 1 i þ 1 i þ þ 2 ð1mÞBi þ 1 1 þ ai þ 1 1 þ ai þ 1 o 1ai þ 1 ð36Þ þ ðai ai þ 1 ÞC i þ 1 Ri

bk ¼

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 C ð1 þ ak Þ ð1mÞ mk sck sk1 þsk s2ck þ dk Ak þ k ð1 þ ak ÞRk1 Gk 2Gk ð37Þ

6. Calculation of annulus’ radii Theoretically, the plastic radii Ri can be obtained using the post-peak rock mass re-rupturing condition, but the equations are

Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 50 (2012) 38–46

seriously nonlinear, which brings out certain inconveniences as follows:

The relative radial stress variable sci at the interface of adjacent

softening annulus is the function of radius Ri, which increases the nonlinear degree and equation order. Under some conditions, the surrounding rock cannot reach the residual state, so the Newton–Raphson method may get a result that Ri oR0 (i¼1,2,y,i0 ), where i0 , smaller than k, is an unknown annulus number. In order to avoid this, the limitation condition that Ri ZR0 should be considered. Finally, for that the material properties of rock mass are predetermined in each annulus, it may obtain that some of the annulus’ plastic shear strain at starting point of a certain ‘step’ is smaller than the predetermined value. Then it would lead to the result that Ri oRi 1, which is obviously wrong. In order to avoid this, the limitation condition of Ri ZRi 1 also should be considered.

Because of the heavy nonlinear, higher equation order and the limitation conditions, the Newton–Raphson method can hardly be carried out in solving the radius Ri of each annulus. Here, we proposed a simple way for obtaining the radius Ri. It was previously mentioned that the stresses of surrounding rock may not satisfy the failure criterion for lower in-situ pressure, hard rock mass and high supporting pressure, and so on, so a criterion should be taken in order to choose appropriate strength parameters of post-peak rock mass during the iterative calculation. It is known that when the softening radius Rk assumed is larger than the actual value, the calculated internal supporting 0 pressure si will be smaller than the actual supporting pressure s0 and the residual radius Ri0 will also be less than R0. Then the corresponding post-peak strength of (i0 þ1)th annulus should be taken as the material properties of the most inner surrounding rock mass, namely material properties of the exposure surface rock mass. The iterative calculation process should be carried out until the error between the calculated support force and actual one is sufﬁciently small. The initial values of upper and lower radii of Rk can be determined by the elasto-brittle–plastic model and ideal elastoplastic model, respectively. And the closed-form solutions can be expressed in the following. The upper and lower softening radii of M–C rock mass can be expressed as h i fk Þ 1=ðNk 1Þ Ruk ¼ R0 ðp þ Dsk0Þð1sin þ Dk ð38Þ h i fk Þ 1=ðN 1 1Þ Rlk ¼ R0 ðp þ Dsk0Þð1sin þD 1

And the upper and lower softening radii of H–B rock mass can be expressed as pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Bc þ B2c 4Ac ðs0 sk Þ Ruk ¼ R0 exp 2Ac pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ð39Þ 2 0 k B þ B p p 4Ap ðs s Þ Rlk ¼ R0 exp 2Ap where Ac ¼m1sc1/4, Bc ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ mk sck s0 þ sk s2ck .

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m1 sc1 s0 þ s1 s2c1 , Ap ¼mksck/4, Bp ¼

The secant method and Newton–Raphson method are combined during the solving process. The ﬂow chart, which summarizes the implementation of the proposed method is shown in following: Step 1: Input geometry and material properties, the number of softening annulus and allowable tolerance.

43

Step 2: Check whether the plastic region exists, if s0 o sk go to Step 3, or exit. Step 3: Calculate the initial value of softening radius Ru, RL and their corresponding calculated supporting pressure s0u and s0L . Then, based on the initial value, update the calculated softening radius Rk based on the secant method. Step 4: Calculate inter radius of each annuls, together with their stress, strain and deformation, Eqs. (11), (13), (21)–(27) for M–C rock mass; Eqs. (17), (19), (28)–(32), for a H–B rock mass. Step 5: Check whether the tolerance between the calculated internal supporting pressure and real one is less than the allowable one, if yes, output the results, or go to Step 3.

7. Example veriﬁcation 7.1. Veriﬁcations for elasto-(brittle)–plastic rock mass In order to validate the results, the typical input data of hard and soft rock mass used by Ogawa and Lo [23] are utilized. The geometric and material properties are presented in Table 1. Dilation angles c ¼01 and 301 are considered, which denote no plastic volume change and plastic volume increase, respectively. In order to guarantee the precision and validate the general applicability, k¼50 is chosen here and the relative error can be analyzed subsequently. In the elasto-brittle–plastic case, b1 should be enough small and 1.0 10 10 is selected. The dimensionless plastic radius (R1/R0) and dimensionless radial displacement uE/(pR0) are calculated and compared with those of analytical results, as shown in Table 2. The results show that the calculation results of two methods are almost the same, and the maximum relative error between this paper and Wang’s results is less than 9.45 10 3% (in the this paper, the relative error between A and B is deﬁned as Abs (A B)/A 100%). 7.2. Veriﬁcations for strain-softening rock mass Alonso et al. [24], Wang et al. [11] and Lee and Pietruszczak [12] presented examples for M–C and H–B softening rock mass, where two sets of parameters are given. Here, they are employed to verify this new approach. The input data of M–C and H–B rock mass are visualized in Figs. 6 and 8, respectively. Several values of ‘k’ are employed to demonstrate the convergence of analytical solutions. In case 2 of M–C rock mass, the dimensionless displacement

Table 1 Geometric and material property parameters. Hard rock

Soft rock

Radius of tunnel, R0 (m) Initial stress, p (MPa) Support pressure, s0 (MPa) Young’s modulus, E (GPa) Poisson’s ratio, v Softening index, bn

1 1 0 50 0.2 1.0 10 10

1 1 0 5 0.2 1.0 10 10

M–C strength parameters ck (MPa) jk (deg.) c1 (MPa) j1 (deg.)

0.173 55 0.061 52

0.276 35 0.055 30

H–B strength parameters mk sk m1 s1 sc1 ¼ sck (MPa)

0.5 1.0 10 4 0.3 1.0 10 5 75

0.2 1.0 10 4 0.05 10 5 50

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Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 50 (2012) 38–46

Table 2 Computed results of relative radius and dimensionless displacement (data in parenthesis are solutions by the methods of [12] and [11]). Calculation scheme

R1/R0

uE/(pR0)

M–C hard rock

ebp(c ¼ 01) ebp(c ¼ 301) ep(c ¼ 01) ep(c ¼ 301)

1.1437(1.1437) 1.1437(1.1437) 1.0585(1.0585) 1.0586(1.0585)

1.5863(1.5863) 2.0799(2.0799) 1.2557(1.2557) 1.3159(1.3159)

ebp(c ¼ 01) ebp(c ¼ 301) ep(c ¼ 01) ep(c ¼ 301)

1.1020(1.1020) 1.1020(1.1020) 1.0603(1.0603) 1.0603(1.0603)

1.4342(1.4342) 1.7094(1.7094) 1.2744(1.2744) 1.3549(1.3549)

M–C soft rock

ebp(c ¼ 01) ebp(c ¼ 301) ep(c ¼ 01) ep(c ¼ 301)

1.7615(1.7615) 1.7615(1.7615) 1.1650(1.1650) 1.1650(1.1650)

4.0442(4.0442) 12.303(12.303) 1.3640(1.3640) 1.5674(1.5674)

H–B soft rock

ebp(c ¼ 01) ebp(c ¼ 301) ep(c ¼ 01) ep(c ¼ 301)

1.6141(1.6141) 1.6141(1.6141) 1.2366(1.2366) 1.2366(1.2366)

3.1870(3.1870) 7.9383(7.9383) 1.5732(1.5732) 2.0768(2.0768)

H–B hard rock

Brittle-plastic method (Wang et al. [11]) and FDM (Lee & Pietruszczk [12])

R =3 m, =0.25, E=10 GPa

σ 0 /p

Rock mass

This paper

p=20.0 Mpa

=15º, =5º c =1.0 MPa, c =0.7 MPa

= =3.75º =8.0e-3

(u0/R0)(2Gk)/(p-σ k) Fig. 6. Ground response curve in strain-softening M–C rock mass.

Table 3 Strain-softening rock mass calculation results.

This paper Rk/R0

2u0Gk/[R0(p sk)]

M–C

Case 1 Case 2 Case 3 Case 4 Wang’s

2.823 2.828 2.854 2.860 2.802

4.099 4.104 4.114 4.123 4.093

28.892 28.961 29.122 29.275 28.813

Case 1 Case 2 Case 3 Case 4 Wang’s

1.963 1.967 1.985 1.988 2.070

2.590 2.593 2.600 2.607 2.638

11.111 11.142 11.224 11.273 12.613

H–B

u/R0/(pR0)/2Gk is 28.961 for k¼50 and that of 29.042 for k¼100. The relative error between condition k¼50 and that of k¼100 is less than 2.80%, so it can be considered to be convergence when k¼ 50. And the same ‘k’ value is used in this paper. Table 3 shows the calculation results with four types of rerupturing conditions and that of numerical results from [11]. It can be seen that the relative error of dimensionless residual radius, dimensionless softening radius and dimensionless displacement at r¼R0 between cases 1 and 4 are 1.31%, 0.59% and 1.33%, and that of the H–B rock mass 1.27%, 0.66% and 1.46%, respectively. So the results with different re-rupturing conditions almost can be considered to be the same when k¼50 is chosen. Furthermore, the results of four types trend to be uniform when k is large enough. Thus the result of case 2 is used to represent the new approach in the following analysis. The relative error of dimensionless displacement at r ¼R0 between analytical method and numerical method is 0.36% with M–C rock mass and that of 11.67% with H–B rock mass, a little larger compared with the error by M–C rock mass. The reason for the difference will be interpreted in the following. Fig. 6 shows the GRC curve of M–C rock mass, and the evolution of the softening and residual radii denoted as Rp and Rs, namely the last and the ﬁrst annulus radii (Rk and R1) respectively, are shown in Fig. 7. The analytical results are in close agreement with those of numerical method [11,12]. In the same way, for the rock mass governed by the H–B criteria, the GRC curve and evolution of Rp and Rs are also calculated, as shown in Figs. 8 and 9, respectively. It is clear that the difference between the two methods is bigger than that of rock mass governed by the M–C criteria, and the difference may be induced by the following two reasons: (1) the heavy nonlinear character of H–B

Brittle-plastic method (Wang et al. [11]) and FDM (Lee & Pietruszczk [12])

σ 0 /p

R1/R0

Rk /R0

R1/R0

R1/R0, Rk/R0 Fig. 7. Evolution of softening and residual radii in strain- softening M–C rock mass.

This paper Brittle-plastic method (Wang et al. [11]) and FDM (Lee & Pietruszczk [12])

/p

Calculation scheme

0

Rock mass

(u0 /R0)(2Gk)/(p-σ k) Fig. 8. Ground response curve in strain-softening H–B rock mass.

criteria; (2) the numerical method is adopted by the mean deformation in each annulus, especially, the tangent stress at Ri is the mean value before and after brittle drop. However, the proposed method in this paper is accurately calculated one by Eq. (17).

Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 50 (2012) 38–46

45

This paper =1.0 =0.7

σ 0 /p

σ 0 /p

Brittle-plastic method (Wang et al. [11]) and FDM (Lee & Pietruszczk [12])

=0.4

Rk/R0

R1/R0

R1/R0, Rk /R0

(u0/R0)(2Gk)/( p-

Fig. 9. Evolution of softening and residual radii in strain- softening H–B rock mass.

Table 4 Computed results of relative radius. Rs/R0

Rp/R0

2u0Gk/[R0(p–sk)]

M–C(l ¼ 1.0) M–C(l ¼ 0.7) M–C(l ¼ 0.4)

2.82822 2.88683 2.98681

4.10363 4.10460 4.10864

28.96137 32.04961 39.56423

H–B(l ¼ 1.0) H–B(l ¼ 0.7) H–B(l ¼ 0.4)

1.96675 1.99417 2.04459

2.59316 2.58682 2.58179

11.14241 12.14120 14.67694

k

)

Fig. 11. GRC curves with different shear modulus deterioration ratios in strainsoftening H–B rock mass.

the two conditions of x1 ¼0.4 and 1.0 are 0.12% and 5.61% using M–C criteria. Additionally, the relative errors of Rp and Rs between the two conditions of x1 ¼0.7 and 1.0 are 2.44% and 1.39%, and the relative errors of Rp and Rs between the two conditions of x1 ¼0.4 and 1.0 are 3.96% and 0.44% using H–B criteria. Thus, the inﬂuence of shear modulus deterioration process on both the residual region and softening region is not signiﬁcant. Whereas, the dimensionless displacement u/[R0(p sk)/2Gk] at r ¼R0 when x1 ¼0.7 and 0.4 increase by 0.66% and 36.61%, respectively, than that of x1 ¼1.0 with M–C criteria, those of 8.96% and 31.72% with H–B criteria. It can be seen that the shear modulus inﬂuences the rock mass deformation heavily, and it can reveal the phenomenon of large deformation in geo-engineering, especially for the deep underground engineering.

1 =1.0

σ 0/p

1 =0.7 1 =0.4

(u0/R0)(2Gk)/(p-

k

)

Fig. 10. GRC curves with different shear modulus deterioration ratios in strainsoftening M–C rock mass.

7.3. Inﬂuence of deformation parameter Here, the inﬂuence of shear modulus deterioration process on circular tunnel surrounding rock deformation is analyzed. Based on Eqs. (33)–(37), the threshold value of x1 can be obtained, namely, 0.39853 and 0.34955 for M–C strain-softening rock mass and H–B strain-softening rock mass, respectively. To analyze the inﬂuence, two another sets of x1 ¼0.7 and 0.4 are selected here. Table 4 shows the calculated results with various x1, and the GRC curves are presented in Figs. 10 and 11 for M–C rock mass and H–B rock mass, respectively. The relative errors of Rp and Rs between the two conditions of x1 ¼0.7 and 1.0 are 0.24% and 2.07%, and the relative errors of Rp and Rs between

8. Conclusions In this study, the closed-form solutions of a circular tunnel excavated in strain-softening M–C and H–B rock mass are obtained, which can be easily degenerated to the classical solutions, such as perfect elasto-plastic and elasto-brittle–plastic rock mass. The post rock mass are divided into a number of annuli, and in each annulus the rock mass has the same material property. Based on the deterioration process of the surrounding rock strength and deformation parameters, the corresponding relationship between rock mass material parameters and plastic shear strain is given, then the re-rupturing conditions, namely the equations for solving each annulus’ radii, are obtained. For the seriously non-linear stresses and displacements of surrounding rock, the combined method of secant method and Newton– Raphson method is used for solving the radii of each annulus. The solution will be more accurate with the increase of annulus number. Based on the fact that failure rock mass deforms easily, the deterioration process is considered. Finally examples are analyzed for validating the correctness, and the results show a good agreement with the former studies. The shear modulus affects the softening and residual radius lightly, but it inﬂuences the rock mass deformation heavily. It should be note that the solutions not only can be used for the strain-softening rock mass but also for the strain-hardening rock mass, even the rock mass including both the hardening and softening process. When the evolution law of rock mass material parameters with the softening index are given, it can be easily solved using the new analytical solutions.

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Acknowledgments This project is supported by the National Natural Science Foundation of China (Project nos. 51174196, 51179185, 51179189, 50804044 and 50804046) and the Scientiﬁc Research Foundation of China University of Mining and Technology (No. 2011RC10). References [1] Park KH, Kim YJ. Analytical solution for a circular opening in an elasticbrittle-plastic rock. Int J Rock Mech Min Sci 2006;43:616–22. [2] Sharan SK. Elastic-brittle–plastic analysis of circular openings in Hoek–Brown media. Int J Rock Mech Min Sci 2003;40:817–24. [3] Hoek E, Carranza-Torres CT, Corkum B. Hoek–Brown failure criterion—2002 edition. In: Proceedings of the ﬁfth North American rock mechanics symposium, Toronto; 2002. p. 267–3. [4] Carranza-Torres C. Elasto-plastic solution of tunnel problems using the generalized form of the Hoek–Brown failure criterion. Int J Rock Mech Min Sci 2004;41:480–1. [5] Sharan SK. Analytical solutions for stresses and displacements around a circular openings in a generalized Hoek–Brown rock. Int J Rock Mech Min Sci 2008;40:78–85. [6] Jiang BS, Zhang Q, He YN, et al. Elastioplastic analysis of cracked surrounding rocks in deep circular openings. Chin J Rock Mech Eng 2007;26:982–6. [7] Park KH, Tontavanich B, Lee JG. A simple procedure for ground response curve of circular tunnel in elastic–strain softening rock masses. Tunnelling Underground Space Technol 2008;23:151–9. [8] Zheng H, Liu DF, Lee CF, Ge XR. Principle of analysis of brittle–plastic rock mass. Int J Solids Struct 2005;42:139–58. [9] Brown ET, Bray JW, Ladanyi B, Hoek E. Ground response curves for rock tunnels. J Geotech Eng ASCE 1983;109(1):15–39.

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