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Analytical solutions for the stresses and deformations of deep tunnels in an elastic-brittle-plastic rock mass considering the damaged zone
Tunnelling and Underground Space Technology 58 (2016) 186–196
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Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Analytical solutions for the stresses and deformations of deep tunnels in an elastic-brittle-plastic rock mass considering the damaged zone Mohammad Reza Zareifard a,⇑, Ahmad Fahimifar b a b
Estahban Institute of Higher Education, Estahban, Iran Amirkabir University of Technology, Tehran, Iran
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 7 March 2015 Received in revised form 16 April 2016 Accepted 23 May 2016
The excavation impact (e.g. due to blasting, TBM drilling, etc.) induces an excavation damaged or disturbed zone around a tunnel. In this regard, in drill and blast method, the damage to the rock mass is more significant. In this zone, the stiffness and strength parameters of the surrounding rock mass are different. The real effect of a damage zone developed by an excavation impact around a tunnel, and its influence on the overall response of the tunnel is of interest to be quantified. In this paper, a fully analytical solution is proposed, for stresses and displacements around a tunnel, excavated in an elastic– brittle–plastic rock material compatible with linear Mohr–Coulomb criterion or a nonlinear Hoek–Brown failure criterion considering the effect of the damaged zone induced by the excavation impact. The initial stress state is assumed to be hydrostatic, and the damaged zone is assumed to have a cylindrical shape with varied parameters; thus, the problem is considered axial-symmetric. The proposed solution is used to explain the behavior of tunnels under different damage conditions. Illustrative examples are given to demonstrate the performance of the proposed method, and also to examine the effect of the damaged zone induced by the excavation impact. The results obtained by the proposed solution indicate that, the effects of the alteration of rock mass properties in the damaged zone may be considerable. Ó 2016 Elsevier Ltd. All rights reserved.
Keywords: Circular tunnel Mohr–Coulomb failure criterion Hoek–Brown failure criterion Brittle-plastic rock Analytical solution Excavation damaged zone
1. Introduction In order to analyze a tunnel, it is essential to understand the various rock mass behaviors after an excavation. The original properties of a rock or rock mass near a tunnel are changed after the excavation. The characteristics of an excavation damaged or disturbed zone (EDZ) vary with the rock mass properties, excavation method, and opening geometry. Such a disturbance can significantly influence the response of the rock mass and the overall performance of the tunnel. Therefore, investigating the influence of the EDZ around an underground excavation is of paramount importance. From this point of view, the characteristics of an EDZ have been extensively investigated for various rock engineering projects, including dam construction, tunnel construction, and waste repository projects. An EDZ can be defined as a rock zone where the rock failure and stiffness parameters have been changed due to the processes related to an excavation. Different mechanisms are related to the
⇑ Corresponding author. E-mail addresses: (A. Fahimifar).
[email protected]
(M.R.
http://dx.doi.org/10.1016/j.tust.2016.05.007 0886-7798/Ó 2016 Elsevier Ltd. All rights reserved.
Zareifard),
[email protected]
development of an EDZ. Major factors related to the development of an EDZ are (a) the excavation impact; and (b) the stress redistribution after the excavation. Based on the factor (a), both TBM and drill and blast excavation disturb the surrounding rock mass. However, in contrast to blasted tunnels, in TBM tunnels, the annular thickness of the EDZ and the rock mass alteration in the EDZ are expected to be insignificant. The extent of the damaged zone induced by excavation impact depends on the rock properties, shape of the tunnel, excavation method and its quality, etc., but it can range from few centimeters in tunnels with TBM to several decimeters and up to several meters with drill and blast (Beackblom and Martin, 1999; Martino and Chandler, 2004). For this reason, the blast-induced damaged zone (BIDZ) has to be considered in the analysis and design of tunnels; while, the TBM drilling-induced damaged rock can be ignored. However, for very high quality controlled blasting, the damage to the rock mass is negligible due to a well-designed blasting pattern and detonation sequence and accurate drilling control. In contrast, the lack of a good blast design and absence of any control on the drilling can result in the significant damage. On the other hand, based on factor (b), due to the stresses induced by the tunnel excavation, a disturbed plastic or fractured zone will develop around the tunnel. However, the elastoplastic
M.R. Zareifard, A. Fahimifar / Tunnelling and Underground Space Technology 58 (2016) 186–196
187
Nomenclature D GSI p0 r Rp ri ur e1 e3 eh er
m rc
disturbance factor Geological Strength Index hydrostatic in situ stress radial distance from the center of the tunnel radius of the plastic zone radius of tunnel radial displacement major principal strain in rock mass minor principal strain in rock mass circumferential strain radial strain Poisson’s ratio of rock mass uniaxial compressive strength of intact rock
response of the tunnel is affected by the damaged zone developed by the excavation impact. It can be concluded that, the factor (b) is influenced by the factor (a). Thus, for an elastoplatic analysis of a tunnel, the effects of the BIDZ developed by the excavation impact must be taken into account. In this regard, one of the significant reasons for assessing the BIDZ around tunnels is its effect on the tunnel stability. This, therefore, implies the need for considering this zone during the tunnel design. Full consideration of the interplay that exists among construction activities, the EDZ, support characteristics, and time requires the use of numerical methods in which all factors can be considered (see, for example Cho et al., 2006; Saiang and Nordlund, 2009a, 2009b; Saiang, 2010). No closed-form solutions exist that include the full complexity of such a problem. Analytical solutions, however, if found, have the potential of providing an insight into the problem, and are very useful to identify the most important variables for a given subject, and contribute to the understanding of the excavation–rock–liner interaction problem. Analytical solutions, however, suffer from distinct limitations, because they usually require a number of assumptions and simplifications that often apply to the problems with limited practical interest. Nevertheless, the advantages of having a closed-form solution often outweigh the limitations. In this regard, no notable analytical solutions that consider the effect of the BIDZ are currently available for analyzing the elastoplastic response of tunnels. In elasto-plastic solutions, proposed by Brown and Bray (1982), Carranza-Torres and Fairhurst (1999), Sharan (2003,2005), Alonso et al. (2003), Park and Kim (2006), Park et al. (2008), Lee and Pietruszczak (2008), Fahimifar and Zareifard (2009, 2012, 2014), Zareifard and Fahimifar (2012, 2014, 2015), Alejano et al. (2010), Cheng (2012), effects of the BIDZ have not been considered, and they are presented for tunnels excavated in initially homogenous conditions. In practice, the global damage induced by blasting is taken into account by using the damage factor D introduced by Hoek et al. (2002). In this method, the blast damage factor D is applied to the entire rock mass surrounding the tunnel. This is a common modeling method which can greatly underestimate the strength and stability of the overall rock mass. In this regard, the blast damage factor D is more appropriate to be applied to the actual zone of damaged rock. In this paper, a fully analytical elastic–brittle–plastic solution for a deep circular tunnel in a Mohr–Coulomb or Hoek–Brown rock mass, considering a cylindrical and homogenous BIDZ is presented. For the models in this paper, it is assumed that the damage induced by blasting is finite in extent and is in the form of a cylindrical zone. Beyond the BIDZ, it is assumed that the rock is not damaged and; therefore, the undamaged rock property values are used.
r1 r3 rr rh
major principal stress minor principal stress radial stress circumferential stress u, C material constants for Mohr–Coulomb rock mass m, s material constants for Hoek-Brown rock mass W dilation angle Subscript ‘D’ refers to quantities corresponding to damaged rock Superscript ‘e’ refers to elastic part of strain Subscript ‘r’ denotes the values of rock mass parameters for plastic zone Subscript ‘i’ denotes the initial values of rock mass parameters Superscript ‘p’ refers to plastic part of strain
The discussion of this paper is restricted to the stable solution, so the topics concerning the instability such as bifurcation and strain localization in the plastic regime (Varas et al., 2005; Alonso et al., 2003) are not considered. In the present exact analytical solution, simple formulas are derived without having to solve complex differential equations. This is useful for acquiring an additional insight into the problem on an opening within a damaged zone; because only a minimal computational effort is needed and considerable economic benefits can be gained by using it in the preliminary stage of a tunnel design. In addition, the exact solution is useful for the verification of the numerical codes and semi-analytical solutions. Furthermore, the proposed solution can also be applied for a tunnel reinforced by means of a grouting zone with increased strength parameters.
2. Definition of the problem The problem considered is shown in Fig. 1. A circular deep tunnel of radius ri is excavated in an initially elastic rock mass characterized by Young’s modulus E, and Poisson’s ratio m. Due to a blasting impact a cylindrical blast-induced damaged zone (BIDZ) will develop around the tunnel with different behavior parameters. In this regard the Young’s modulus and the Poisson’s ratio of BIDZ are ED and mD, respectively. Axial symmetry conditions for geometry and loading can be assumed for the problem of the tunnel under a uniform initial stress p0. A uniform internal pressure ri ¼ rrðri Þ is considered to act on the periphery of the tunnel as a result of a lining installation. The stress redistribution and displacements will take place, due to excavation of the tunnel, installing the lining and the alteration of the rock mass in the BIDZ. As ri is gradually reduced, a radial displacement occurs and a plastic zone develops around the tunnel as ri becomes less than the initial yield stress. After failure, the strength of the rock suddenly drops and follows the post-failure softening behavior. In this study, both the damaged and undamaged rock masses are considered to be elastic–brittle-plastic (in special case: perfectly plastic) as shown in Fig. 2. Two different zones may forms around the tunnel: an external elastic zone and an internal plastic zone of radius Rp. In this respect, three different cases can be considered, depending on the extent of the BIDZ and the plastic zone (Fig. 1): Case (a): the radius of the plastic zone is larger than the BIDZ. Case (b): the radius of the BIDZ is greater than the plastic zone. Case (c): the radius of the BIDZ is equal to the radius of the plastic zone.
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r
BIDZ
Plastic zone
r r
r Elastic Zone
i
RD
ri
At infinity
p0
r
Rp
(a)
r
Plastic zone
r r
r
BIDZ
Elastic Zone i
ri
At infinity
r
p0
drrðrÞ rhðrÞ rrðrÞ ¼0 dr r
Rp
RD
The analyses for three cases are different. In this way, for analysis of a tunnel with an unknown plastic radius, the calculations are, firstly, carried out based on the case (a) as a choice. Then, if the calculated plastic radius becomes larger than the radius of BIDZ; the chosen case is correct, and the results are acceptable. Otherwise, the case (b) should be taken into account. Now, if the calculated plastic radius is smaller than the radius of the BIDZ, the results are acceptable. Otherwise, the case (c) must be considered. In the proposed solution, the effect of the gravitational loadings in the plastic zone is ignored. However, in highly fractured and damaged rock masses, as shown by Carranza-Torres and Fairhurst (1997) and Zareifard and Fahimifar (2012), the weight of the plastic zone may affect the response of the rock mass. Considering the axial-symmetry condition of the problem in Fig. 1 (three cases), the resulting stress state at a distance r is defined by the radial stress rr(r) and circumferential stress rh(r) which are the minor and major principal stresses, r3(r) and r1(r), respectively. The resulting displacement is defined by the radial displacement ur(r). It should be noted that for such long tunnel problem, the out-of-plane stress rz(r), can be taken as the intermediate principal stress r2(r). For the considering axial-symmetric problem, the equilibrium equation relating the radial stress rr(r) and the circumferential stress rh(r) at the radial distance r is (Timoshenko and Goodier, 1982):
The radial strain er(r) and circumferential strain eh(r) at radius r for axial-symmetric condition can be stated in terms of the radial displacement ur(r), as follows (Timoshenko and Goodier, 1982):
(b)
eh ¼ r
r r
r
Plastic zone
Elastic Zone
BIDZ i
ri
At infinity
r
Rp
ð1Þ
p0
RD
(c) Fig. 1. The circular deep tunnel subjected to a hydrostatic stress field with a cylindrical BIDZ: (a) Rp > RD, (b) Rp < RD, and (c) Rp = RD.
ur ; r
er ¼
dur dr
ð2Þ
For both rock masses (the BIDZ and undamaged zone), the induced elastic strains eer and eeh in terms of the final stresses rr and rh with consideration of the initial hydrostatic stress p0, can be written as (Timoshenko and Goodier, 1982):
1þm ½ð1 mÞðrr p0 Þ mðrh p0 Þ E 1þm ¼ ½ð1 mÞðrh p0 Þ þ mðrr p0 Þ E
eer ¼
ð3Þ
eeh
ð4Þ
where E and m are Young’s modulus and Poisson’s ratio of the rock mass, respectively. In this regard, rock mass parameters for the BIDZ are denoted by a subscript ‘D’; namely, the elastic parameters for the BIDZ are ED and mD.
Fig. 2. Behavior model used in this study for both the damaged and undamaged rock masses: (a) Stress-strain relationship. (b) Flow rule.
M.R. Zareifard, A. Fahimifar / Tunnelling and Underground Space Technology 58 (2016) 186–196
For the plastic regime, two most commonly used failure criteria are considered: the nonlinear Hoek-Brown failure criterion (Hoek et al., 2002):
r1 r3 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mr3 rc þ sr2c
ð5-1Þ
and the linear Mohr–Coulomb failure criterion,
r1 ¼ A þ Br3
ð5-2Þ
cos u u ; r1 and r3 are the major and minor where A ¼ 2C ; B ¼ 1þsin 1sin u 1sin u
principal stresses at failure, respectively; rc is the uniaxial compressive strength of the intact rock material; m and s are material constants for the Hoek-Brown failure criterion; and C and u are material constants for the Mohr–Coulomb failure criterion (C and u are the cohesion and the friction angle of the rock, respectively). For the present axial-symmetry solution, principal stresses, r1 and r3 are equal to the radial and circumferential stresses rr and rh, respectively. Eq. (5) may thus be rewritten as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rh rr ¼ mrr rc þ sr2c rh ¼ A þ Brr
ð6-1Þ ð6-2Þ
rh rr r h ¼ Ai þ B i r r
ð7-1Þ ð7-2Þ
ui i cos ur ; and Ci, ui and mi, si are material where Ai ¼ 2C ; Bi ¼ 1þsin 1sin u 1sin u i
i
constants for the initial rock mass. And for the plastic zone r i 6 r 6 Rp (Fig. 1), Eq. (6) is modified as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rh rr ¼ mr rr rc þ sr r2c rh ¼ Ar þ Br rr
ð8-1Þ ð8-2Þ
ur r cos ur ; and Cr, ur and mr, sr are material where Ar ¼ 2C ; Br ¼ 1þsin 1sin u 1sin u r
r
constants for the failed rock mass. The failure parameters Ci, ui and mi, si and the elastic parameter E vary with rock type and rock mass quality as measured by is the Geological Strength Index of the rock mass (GSI) (Hoek et al., 2002). In the broken or plastic zone, the reduced strength parameters Cr, ur and mr, sr can be obtained using the residual GSI according to a method proposed by Alejano et al. (2009). It should be noted that the damaged rock mass has different failure parameters, here denoted by subscript ‘D’ (namely, rcD, CiD, uiD, miD, siD, CrD,urD, mrD, srD). These parameters can be attained considering a suitable damage factor (D) (Hoek et al., 2002). In this regard, the blast damage factor was first introduced in the 2002 version of the Hoek-Brown criterion (Hoek et al., 2002) and it is used for the estimation of the strength and deformability parameters. In the plastic zone, the total radial and circumferential strains, er and eh, can be divided into the elastic and plastic parts:
er ¼ eer þ epr ; eh ¼ eeh þ eph
ð9Þ
By assuming that the elastic strains are relatively small as compared to the plastic strains and that a non-associated flow rule is valid, the plastic parts of radial and circumferential strains may be related as (Sharan, 2003; Park and Kim, 2006):
epr þ K W eph ¼ 0
Most of the researchers who have studied dilatancy in rock and rock masses have acknowledged the need to use non-associative flow rules. Given that the performance of significant tests on rock masses is a complex matter, dilatancy studies should be based on field experiments and back-analysis (Alejano and Alonso, 2005). In this regard, the analytical solutions like the present closedform solution can be beneficial. Based on wide practical rock engineering experience and probably inspired by the idea of providing the rock mechanics community with parameters for feeding numerical models, Hoek and Brown (1997) recommended the use of constant dilatancy angle values based on rock mass quality; they proposed, thus, values of W = u/4 for excellent- and goodquality rock masses, W = u/8 for average-quality rock masses, and W = 0 for poor-quality rock masses. From Eqs. (2) and (10), the displacement compatibility equation can be expressed as (Park and Kim, 2006):
dur ur þ K W ¼ f ðrÞ dr r
ð11Þ
where
f ðrÞ ¼ B1 þ B2 rrðrÞ þ B3 rhðrÞ
ð12Þ
B1 ¼ ð1þmÞð12EmÞð1þK W Þ p0
For the initial rock mass, Eq. (6) can be written as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ mi rr rc þ si r2c
189
ð10Þ
where KW = (1 + sin W)/(1 – sin W), and W is the dilation angle of the rock mass (in the damaged zone the dilation angle is shown by WD).
B2 ¼ ð1þmÞð1E mK W mÞ
B3 ¼ ð1þmÞðK WEmK W mÞ It should be noted that, the damaged rock mass has altered parameters. Thus for the damaged zone the following coefficients are used:
B1D ¼ ð1þmD Þð12EDmD Þð1þK WD Þ p0 B2D ¼ B3D ¼
ð1þmD Þð1mD K W
m Þ D D
ED ð1þmD ÞðK WD mD K W
m Þ D D
ED
3. Analysis of the elastic zone As shown in Fig. 1, outside the plastic zone the rock mass remains elastic. The solutions are different for the cases of (a), (b) and (c). 3.1. Cases (a) and (c) An external elastic zone of the inner radius Rp with unaltered parameters exists outside the plastic zone. The stresses and radial displacements in this zone, can be derived based on the thickwalled cylinder theory in Elasticity (Timoshenko and Goodier, 1982):
R2p r2 R2p rhðrÞ ¼ p0 þ ðp0 rrðRp Þ Þ 2 r R2p 1þm ðp0 rrðRp Þ Þ urðrÞ ¼ E r
rrðrÞ ¼ p0 ðp0 rrðRp Þ Þ
ð13Þ ð14Þ ð15Þ
3.2. Case (b) The elastic zone consists of an external undamaged zone with unaltered parameters and an internal BIDZ. As shown in Fig. 1b, the undamaged and damaged elastic zones interact with each other at the radius RD with a radial stress rrðRD Þ .
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The stresses and displacements in the undamaged zone and BIDZ are obtained using the thick-walled cylinder in Elasticity:
R2D r2 R2D rhðrÞ ¼ p0 þ p0 rrðRD Þ 2 r R2D 1 þ m urðrÞ ¼ p0 rrðRD Þ E r
rrðrÞ ¼ p0 p0 rrðRD Þ
ð16Þ ð17Þ ð18Þ
for the undamaged zone, and
rrðrÞ ¼ rrðRD Þ rrðRp Þ
rhðrÞ ¼ rrðRD Þ rrðRp Þ
R2p
R2p
R2 1 2D r
R2D R2p
R2D r2
R2D
R2p
1þ
! þ rrðRD Þ
ð19Þ
þ rrðRD Þ
ð20Þ
!
R 1 þ mD R2 ¼ rrðRD Þ rrðRp Þ 2 p 2 ð1 2mD Þ þ D ED r RD Rp 2
urðrÞ
!
1 þ mD þ ðrrðRD Þ p0 Þð1 2mD Þr ED
a1 ¼
m
2 DÞ
ðED ð1 þ mÞðR2D R2p Þ Eð1 þ mD Þðð2mD 1ÞR2D R2p ÞÞ
ð26-1Þ
for Mohr–Coulomb failure criterion and,
rrðrÞ ¼ ri þ MD log
r 2 r þ N D log ri ri
ð26-2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! MD ¼ mrD ri rcD þ sr r2cD . ND ¼ mrD4rcD Then, solving the differential equation for undamaged zone leads to the following equation: for Hoek-Brown failure criterion, where
rrðrÞ ¼
!, 1BrD RD ð1 Br Þ Ar þ ð1 Br ÞrrðRD Þ þ Ar r
ð27-1Þ
for Mohr–Coulomb failure criterion and,
r r 2 þ Nlog RD RD
ð27-2Þ
ð21Þ
ð22Þ
2R2p Eð1
r 1BrD i ðð1 BrD Þri þ ArD Þ ð1 BrD Þ r
rrðrÞ ¼ rrðRD Þ þ M log
for the BIDZ. The boundary pressure rrðRD Þ between these elastic zones is obtained by using the equilibrium and compatibility conditions at the boundary between them, i.e. at radius RD.
rrðRD Þ ¼ a1 ðrrðRD Þ p0 Þ þ p0
rrðrÞ ¼ ArD þ
ð23Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M ¼ mr rrðRD Þ rc þ sr r2c . mr rc N¼ 4 In Eqs. (27), the radial stress rrðRD Þ is obtained using Eqs. (26). The corresponding value of the circumferential stress rh,is calculated using the failure criterion (i.e. Eqs. (8)). The radius of the plastic zone Rp can be obtained by considering the continuity of the radial stress at the elastic–plastic interface. Consequently, the radial stress obtained at the plastic radius from Eqs. (27) must be equal to the corresponding value in Eq. (25); because, the radial stress must be continues over the boundary. Thus, the following equation for the radius of the plastic zone, Rp is derived: for Hoek-Brown failure criterion, where
Rp ¼ RD
4. Analysis of the plastic zone
0
4.1.1. Case (a) The plastic zone consists of an external undamaged zone with unaltered parameters and an internal damaged zone (Fig. 1a). As shown in Fig. 1a, the undamaged and damaged zones interact with each other at the radius RD with a radial stress rrðRD Þ . The elastic and plastic zones interact with each other at the plastic radius Rp, where the radial boundary pressure rrðRp Þ is applied. Utilizing Eq. (14), the circumferential stress rhðRp Þ at the radius Rp, is obtained as:
rhðRp Þ ¼ 2p0 rrðRp Þ
ð24Þ
rhðRp Þ and rrðRp Þ must satisfy the failure criterion for the undamaged rock; therefore, substituting these stresses into Eq. (7), and solving the derived equation, gives the boundary radial stress,
rrðRp Þ ¼ ð2p0 Ai Þ=ðBi þ 1Þ
ð25-1Þ
for Mohr–Coulomb failure criterion and,
rrðRp Þ
ð28-1Þ
for Mohr–Coulomb failure criterion and,
4.1. Stresses
1 ¼ p0 þ rc mi 8
1 Ar rrðRp Þ ð1 Br Þ 1Br Ar rrðRD Þ ð1 Br Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 16mi p0 þ m2i þ 16si
rc
ð25-2Þ
for Hoek-Brown failure criterion. After obtaining the value of rrðRp Þ from Eq. (25), the corresponding value of rhðRp Þ is obtained from Eq. (24) or Eq. (7). Then, the radial displacement urðRp Þ is obtained using Eq. (15). By substituting Eq. (8) into Eq. (1) and solving the derived differential equation by using the boundary condition rrðri Þ ¼ ri at r = ri, the following closed-form equation for the radial stress rr(r) in the damaged area of the plastic zone ðr 6 RD Þ is obtained:
Rp ¼ RD exp @
M þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 M 2 4NðrrðRD Þ rrðRp Þ Þ A 2N
ð28-2Þ
for Hoek-Brown failure criterion 4.1.2. Case (b) The elastic zone and the damaged plastic zone interact with each other at the plastic radius Rp, where the radial boundary pressure rrðRp Þ is applied. Utilizing Eq. (20) by considering Eq. (22), the circumferential stress rhðRp Þ at the radius Rp, is obtained as:
rhðRp Þ ¼ ð1 þ aÞp0 arrðRp Þ
ð29Þ
a ¼ a2 a1 a1 a2 where a1 is obtained from Eq. (23) and,
a2 ¼
R2D þ R2p R2D R2p
rhðRp Þ and rrðRp Þ must satisfy the failure criterion for the damaged rock; therefore, substituting these stresses into Eq. (7), and solving the derived equation, gives the boundary radial stress,
rrðRp Þ ¼
ða þ 1Þp0 AiD BiD þ a
ð30-1Þ
for Mohr–Coulomb failure criterion and, 0
rrðRp Þ ¼ p0 þ
1
r @mi 2 cD
2ða þ 1Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4ða þ 1Þ2 miD p0 þ m2iD þ 4ða þ 1Þ2 siD A
rcD
ð30-2Þ for Hoek-Brown failure criterion
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M.R. Zareifard, A. Fahimifar / Tunnelling and Underground Space Technology 58 (2016) 186–196
After obtaining the value of rrðRp Þ from Eq. (30), the corresponding value of rhðRp Þ is obtained from Eqs. (29) or (7). Then, the radial displacement urðRp Þ is obtained using Eq. (21). Solving Eq. (1) for the failed rock mass (using Eq. (8)) applying the boundary condition rrðri Þ ¼ ri at r = ri, the radial stress rr(r) in the damaged plastic zone is obtained from Eq. (26). The corresponding value of the circumferential stress rh, is calculated using the failure criterion (i.e. Eqs. (8)). The radial stress obtained at the plastic radius from Eqs. (26) must be equal to the corresponding value in Eq. (30); because, the radial stress must continue over the boundary. Thus, the following equation for the radius of the plastic zone, Rp is derived,
1 ArD rrðRp Þ ð1 BrD Þ 1BrD Rp ¼ r i ArD ri ð1 BrD Þ
On the other hand, solving the differential Eq. (11) for the damaged area ðr 6 RD Þ gives:
ur ¼
1 rK WD
ðFðrÞ FðRD ÞÞ þ urðRD Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 MD þ M 2D 4ND ðri rrðRp Þ Þ A Rp ¼ r i exp @ 2ND
rD ÞðB2D þB3D BrD Þ 3D BrD Þ B1D þ B3D ArD ArD ðBðB2DrDþB þ ðri ðBrDðK1ÞþA r ðK WD þ1Þ 1Þ W þBrD ÞðBrD 1Þ
FðrÞ ¼
D
ðK WD þ 1Þ
and for Hoek-Brown failure criterion:
0 B @
FðrÞ ¼ r ðK WD þ1Þ
1 2 ðK WD þ 1Þ2 B4D þ B5D log rri þ B6D log rri C A r ðK WD þ 1Þ 2B6D log r þ B5D þ 2B6D i
B4D ¼ B1D þ B2D ri þ B3D ðri þ MD Þ ð31-2Þ
for Hoek-Brown failure criterion In case (b), since, the radial stress at the plastic radius obtained from Eq. (30) is a function of the plastic radius, Rp, the solution is carried out by iteration. A trial can be attempted with an assumed value of Rp, then, the values of the radial stress and the plastic radius are calculated from Eqs. (30) and (31), alternately, to achieve a certain convergence.
ð33Þ
where for Mohr–Coulomb failure criterion,
ð31-1Þ
for Mohr–Coulomb failure criterion and,
K W D RD r
ðK W þ1Þ3 D
B5D ¼ B2D MD þ B3D ð2ND þ M D Þ B6D ¼ ðB2D þ B3D ÞND 4.2.2. Cases (b) and (c) Solving the differential Eq. (11) for the damaged plastic zone ðr 6 Rp Þ gives:
ur ¼
1 rK WD
ðHðrÞ HðRp ÞÞ þ urðRp Þ
K W D Rp r
where for Mohr–Coulomb failure criterion, 4.1.3. Case (c) In this case, the plastic radius is known, RD = Rp. The elastic zone and the damaged plastic zone interact with each other at the plastic radius RD = Rp, where the radial boundary pressure rrðRp Þ is applied. Solving Eq. (1) for the failed rock mass (considering Eq. (8)) using the boundary condition rrðri Þ ¼ ri at r = ri, the radial stress rr(r) in the damaged plastic zone is obtained from Eq. (26). The corresponding value of the circumferential stress rh, is calculated using the failure criterion (i.e. Eqs. (8)). For the plastic radius, obtaining the value of rrðRp Þ from Eq. (26), the corresponding value of rhðRp Þ is obtained from Eq. (8). Then, the radial displacement urðRp Þ is obtained using Eq. (15). 4.2. Displacements 4.2.1. Case (a) Solving the differential Eq. (11) by using the boundary radial displacement at the plastic radius urðRp Þ , the following closedform equation for the radial displacement ur(r) in the undamaged area of the plastic zone ðRD 6 r 6 Rp Þ is obtained:
ur ¼
1 rKW
ðGðrÞ GðRp ÞÞ þ urðRp Þ
K W Rp r
where for Mohr–Coulomb failure criterion:
GðrÞ ¼
ðrrðR Þ ðBr 1ÞþAr ÞðB2 þB3 Br Þ 3 Br Þ D B1 þ B3 Ar Ar ðBðB2rþB þ rðK W þ1Þ ðK W þBr ÞðBr 1Þ 1Þ ðK W þ 1Þ
and for Hoek-Brown failure criterion:
1 2 ðK W þ 1Þ2 B4 þ B5 log RrD þ B6 log RrD @ A ðK W þ 1Þ 2B6 log RrD þ B5 þ 2B6 0
GðrÞ ¼ rðK W þ1Þ
ðK W þ1Þ3
B4 ¼ B1 þ B2 rrðRD Þ þ B3 ðrrðRD Þ þ MÞ B5 ¼ B2 M þ B3 ð2N þ MÞ B6 ¼ ðB2 þ B3 ÞN
ð32Þ
HðrÞ ¼
rD ÞðB2D þB3D BrD Þ 3D BrD Þ B1D þ B3D ArD ArD ðBðB2DrDþB þ ðri ðBrDðK1ÞþA rðK WD þ1Þ 1Þ W þBrD ÞðBrD 1Þ D
ðK WD þ 1Þ ð40-1Þ
and for Hoek-Brown failure criterion:
0 B @
HðrÞ ¼ r ðK WD þ1Þ
1 2 ðK WD þ 1Þ2 B4D þ B5D log rr þ B6D log rr i i C A r ðK WD þ 1Þ 2B6D log ri þ B5D þ 2B6D
B4D ¼ B1D þ B2D ri þ B3D ðri þ MD Þ
ðK W þ1Þ3 D
B5D ¼ B2D MD þ B3D ð2ND þ M D Þ B6D ¼ ðB2D þ B3D ÞND 5. Practical applications In order to investigate the effect of the BIDZ on the displacement and stresses, the results obtained from the solution for cases of hard and soft rocks, are compared using different disturbance conditions. Here, properties of rocks and data for underground openings are taken from the published sources (Carranza-Torres and Fairhurst, 1999; Alejano et al., 2009). 5.1. A tunnel in a soft elastic–perfectly plastic rock mass Here, the proposed solution is used for parametric study to observe the effects of the BIDZ on the overall response of a tunnel in a soft elastic–perfectly plastic rock mass. The initial rock mass parameters used in the failure criteria can be estimated by determining the unconfined compressive strength of intact rock samples rc, the Geological Strength Index (GSI), the rock parameter mintact (which is a Hoek-Brown material constant for intact samples), and the Disturbance factor (D). The Rocscience program RocLab is capable of estimating rc and mintact by either fitting laboratory data or by using built-in charts of typical parameter
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ranges. The GSI (Hoek et al., 2002) is a parameter that is used to help link the mi and si parameters to field observations of the rock mass. The disturbance factor D, which varies from zero (e.g., an excellent quality controlled blasting or excavation) to one (e.g., a significant disturbance due to heavy production blasting), is used to capture damage due to blasting or disturbance. One of the data sets appearing in (Carranza-Torres and Fairhurst, 1999; Sharan, 2003) is taken as this input data. An intact rock core, sampled from the rock mass and tested in the laboratory, indicates the following (intact rock) properties:
rc ¼ 30 MPa; mintact ¼ 10; v ¼ 0:25 The rock mass has a GSI value of 50. All of these parameters are used to determine mi, si, ui and Ci respectively. In this way, considering the elastic–perfectly plastic behavior, the ‘scaling’ failure parameters in Mohr-Coulomb and Hoek-Brown failure criteria for the undamaged rock mass can be obtained using RocLab (Rocscience, 2002) considering D = 0. Hoek-Brown parameters:
mi ¼ mr ¼ 1:7;
si ¼ sr ¼ 0:0039;
rc ¼ 30 MPa:
miD ¼ mrD ¼ 0:8;
siD ¼ srD ¼ 0:001;
rcD ¼ 30 MPa:
Mohr-Coulomb parameters:
uiD ¼ urD ¼ 20 ; C iD ¼ C rD ¼ 1:5 MPa: These values are obtained using the damage factor of D = 0.6 and similar GSI and intact rock material properties with undamaged rock. In addition, Poisson’s ratio assumed to be unaltered for the damaged zone, in contrast to the initial rock mass, i.e. vD = 0.25. A supported circular tunnel of radius ri = 5 m, excavated in a rock mass having the mechanical properties described above, is considered. The far-field stress is assumed to be p0 = 30 MPa and the tunnel is supported by an internal pressure of ri ¼ 5 MPa. In order to investigate the effect of the thickness of the BIDZ on the stresses and displacements in the rock mass, the results are compared for different radial extents of the BIDZ. The dimensionless radial displacement, urir ½% and the dimensionless stresses rrh and rrr , in the rock mass are plotted in i
Mohr-Coulomb parameters:
ui ¼ ur ¼ 26 ; C i ¼ C r ¼ 2:1 MPa: On the other hand, for the rock mass considered in this example (GSI = 50 and rc ¼ 30 MPa), RocLab indicates a value of E ¼ 5500 MPa. Both damaged and undamaged rock masses obey a nonassociated flow rule with no plastic dilation, i.e. W = WD = 0. It should be notified that, in the proposed models, it is assumed that, the disturbance is in the form of the blast induced-damaged zone around the tunnel boundary. Here, for estimating the rock mass parameters in the damaged zone the notion of disturbance factors, D, (Hoek et al., 2002) is used. It is assumed that the damaged rock mass is characterized by a D value of 0.6; thus, using RocLab (Rocscience, 2002), the rock mass parameters for the blast-induced damaged zone are obtained as: Young’s modulus:
ED ¼ 3800 MPa
Fig. 3. Dimensionless radial displacement,
Hoek-Brown parameters:
ur ri
i
Figs. 3–6 for cases RD = 5 m, RD = 7 m, RD = 10.5 m and RD = 15 m. While the high values chosen for the extent of the BIDZ may not be usually encountered in practice, choosing these high values, e.g. RD = 15 in this study is to investigate the maximum possible effect of BIDZ thickness on the predicted displacements and stresses, and to illustrate cases (b) and (c) (see Fig. 1(b and c)). For RD = 5m the rock mass remains totally undamaged. On the other hand, for RD = 7 m, RD = 10.5 m and RD = 15 m, the results show that cases (a), (c) and (b) should be used, respectively. It is observed that, the radius of the damaged zone and degree of disturbance in this zone are the most important parameters. However, measuring these parameters is difficult, especially in the early design of tunnels. Hence, incorporating the damaged zone in the early design stage of tunnels is often neglected. One of the main advantages of the proposed solution is that, in contrast to standard FDM or FEM-based codes, it permits performing parametric studies more conveniently. Parametric studies are usually very recommendable in such cases where a certain deal of uncertainty is always expected. This kind of analysis is very helpful for
½% (for soft rock): (a) Hoek-Brown rock mass and (b) Mohr-Coulomb rock mass.
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193
Fig. 4. Dimensionless circumferential stresses rrhi (for soft rock): (a) Hoek-Brown rock mass and (b) Mohr-Coulomb rock mass.
Fig. 5. Dimensionless radial stresses rrri (for soft rock): (a) Hoek-Brown rock mass and (b) Mohr-Coulomb rock mass.
controlling uncertainties encountered in the quantification of the damaged zone. In this way, Figs. 6 and 7 illustrate the effect of the disturbance factor (D) and the radius of the damaged zone on the radius of the plastic zone and the tunnel convergence, respectively. The corners observed in the diagrams are because of the fact that at these points the radii of the damaged zone become equal to the radii of the plastic zone (i.e. transfer to another case). The results show that, if a wide damaged zone with high reduced parameters develops around the tunnel, its influence is significant on the stresses and displacements around the tunnel
(i.e. for RD > 1.5ri and D P 0.5). In Figs. 3–7, the results for both Hoek-Brown and Mohr-Coulomb failure criteria are plotted. The difference in the results of those failure criteria seems insignificant, especially when RD is low.
5.2. A tunnel in a hard elastic–brittle–plastic rock mass Now, the effects of the BIDZ on the response of a tunnel in a hard Hoek–Brown elastic–brittle–plastic rock mass are investigated.
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Fig. 6. Influence of disturbance factor (D) and radius of the damaged zone (RD) on radius of the plastic zone Rp (for soft rock): (a) Hoek-Brown rock mass and (b) MohrCoulomb rock mass.
Fig. 7. Influence of disturbance factor (D) and the radius of the damaged zone (RD) on tunnel convergence urðri Þ (for soft rock): (a) Hoek-Brown rock mass and (b) Mohr-Coulomb rock mass.
One of the data sets appearing in (Alejano et al., 2009) is taken as this input data. An unsupported circular tunnel of radius ri = 7 m is excavated in a rock mass (GSI = 64.9) composed of conglomerate, sandstone and mudstone to a depth of 1000 m, where an in situ hydrostatic stress p0 = 27 MPa is applied. Laboratory testing indicated values for rc = 162 MPa, mintact = 19, v = 0.25. These parameters are used to determine Young’s modulus and the initial Hoek-Brown parameters for the undamaged rock mass, E, mi, si, using RocLab (Rocscience, 2002) considering D = 0:
mi ¼ 5:4;
si ¼ 0:02;
E ¼ 24 GPa:
For estimating these parameters in the damaged zone, it is assumed that the disturbed rock mass is characterized by a D value of 0.6:
miD ¼ 3:2;
siD ¼ 0:0076;
ED ¼ 16 GPa:
It is assumed that, in the damaged zone the parameters rc and v remain unchanged. According to a method proposed by Alejano et al. (2009), the residual GSI is 27.8, and the corresponding residual
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Fig. 8. Dimensionless radial displacement,
ur ri
½% for the tunnel in hard rock.
Fig. 10. Dimensionless radial stresses rrr for the tunnel in hard rock. i
Fig. 9. Dimensionless circumferential stresses
rh ri
for the tunnel in hard rock.
Hoek-Brown strength parameters are mr = 1.4, mrD = 0.5, sr = 0.0003, srD = 0.00004. Both damaged and undamaged rock masses obey a nonassociated flow rule with dilatancy angles W ¼ WD ¼ 15 . In order to investigate the effect of the thickness of the BIDZ on the stresses and displacements in the rock mass, the results are compared for different radial extents of the BIDZ. The dimensionless radial displacement, urir ½% and the dimensionless circumferential stress rrh , in the rock mass are plotted in i
Fig. 11. Influence of disturbance factor (D) and radius of the damaged zone (RD) on radius of the plastic zone Rp for the tunnel in hard rock.
Figs. 8–10 for cases RD = 7, RD = 8, RD = 9.5 and RD = 12. For RD = 7 m the rock mass remains totally undamaged. On the other hand, for RD = 8, RD = 9.5 and RD = 12, the results show that case (a), (c) and (b) must be used, respectively. Figs. 11 and 12 illustrates the effect of the disturbance factor (D) and the radius of the damaged zone on the radius of the plastic zone and the tunnel convergence, respectively. As observed, in the hard rock similar results are obtained, namely for RD > 1.5ri and D P 0.5, the effect of the damaged zone is not negligible.
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Fig. 12. Influence of disturbance factor (D) and the radius of the damaged zone (RD) on tunnel convergence urðri Þ for the tunnel in hard rock.
6. Conclusions In this work, simple closed-form analytical solutions of the displacements and stresses for the elastic–brittle–plastic analysis of a circular tunnel excavated in Hoek–Brown or Mohr–Coulomb rock mass were introduced in a theoretically consistent way. In the proposed method, the effect of a homogeneous and isotropic damaged zone induced by the excavation impact (mainly, due to blasting), around the tunnel was taken into account. Since the radius of the plastic zone is unknown initially; three different cases were introduced depending on the radii of the blastinduced damaged and the plastic zones. The solutions for these cases were derived, and the results were compared for tunnels in soft and hard rock masses. The results, obtained by using the proposed solution, indicate that a blast-induced damaged zone has a significant effect on the stresses and displacements in the rock mass, when the disturbance and the radius of the damaged zone are relatively high. References Alejano, L.R., Alonso, E., 2005. Considerations of the dilatancy angle in rocks and rock masses. Int. J. Rock Mech. Min. Sci. 42 (4), 481–507. Alejano, L.R., Alonso, E., Dono, A.R., Feranndez-Man, G., 2009. Ground reaction curves for tunnels excavated in different quality rock masses showing several types of post-failure behaviour. Tunn. Undergr. Space Technol. 24, 689–705. Alejano, L.R., Alonso, E., Dono, A.R., Feranndez-Man, 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.
Alonso, E., Alejano, L.R., Varas, F., Fdez-Manin, G., Carranza-Torres, C., 2003. Ground response curves for rock masses exhibiting strain softening behavior. Int. J. Numer. Anal. Meth. Geomech. 27, 1153–1185. Beackblom, G., Martin, C.D., 1999. Recent experiments in hard rocks to study the excavation response: implications for the performance of a nuclear waste geological repository. Tunn. Undergr. Space Technol. 14 (3), 377–394. Brown, E.T., Bray, J.W., 1982. Rock-support interaction calculations for pressure shafts and tunnels. In: ISRM Symposium, Aachen, pp. 555–565. Carranza-Torres, C., Fairhurst, C., 1997. On the stability of tunnels under gravity loading, with post-peak softening of the ground. Int. J. Rock Mech. Min. Sci. 34, 3–4. 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. Cho, K.-H., Park, B.-K., Lee, I.-M., 2006. Blast-induced damage analysis in anisotropic rock condition. Tunn. Undergr. Space Technol. 21, 368. Cheng, Y.M., 2012. Modified Kastner formula for cylindrical cavity contraction in Mohr–Coulomb medium for circular tunnel in isotropic medium. J. Mech. 28 (1), 559–565. Fahimifar, A., Zareifard, M.R., 2009. A theoretical solution for analysis of tunnels below groundwater considering the hydraulic–mechanical coupling. Tunn. Undergr. Space Technol. 24, 634–646. Fahimifar, A., Zareifard, M.R., 2012. A new closed-form solution for analysis of unlined pressure tunnels under seepage forces. Int. J. Numer. Anal. Meth. Geomech. 37 (11), 1591–1613. Fahimifar, A., Zareifard, M.R., 2014. A new elasto-plastic solution for analysis of underwater tunnels considering strain-dependent permeability. Struct. Infrastruct. Eng.: Maint. Manage. Life-Cycle Des. Perform. 10 (11), 1432–1450. Hoek, E., Brown, E.T., 1997. Practical estimates of rock mass strength. Int. J. Rock Mech. Sci. Geomech. Abstr. 34 (8), 1165–1187. Hoek, E., Carranza-Torres, C.T., Corkum, B., 2002. Hoek–Brown failure criterion – 2002 edition. In: Proceedings of the 5th North American Rock Mechanics Symposium and 17th Tunnelling Association of Canada Conference, Toronto, pp. 267–273. Lee, Y.K., Pietruszczak, S., 2008. A new numerical procedure for elasto-plastic analysis of a circular opening excavated in a strain-softening rock mass. Tunn. Undergr. Space Technol. 23 (5), 588–599. Martino, J.B., Chandler, N.A., 2004. Excavation-induced damage studies at the underground research laboratory. Int. J. Rock Mech. Min. Sci. 41, 1413–1426. Park, K.-H., Kim, Y.-J., 2006. Analytical solution for a circular opening in an elastic– brittle–plastic rock. Int. J. Rock Mech. Min. Sci. 43, 616–622. Park, K.-H., Tontavanich, B., Lee, J.-G., 2008. A simple procedure for ground response curve of circular tunnel in elastic-strain softening rock masses. Tunn. Undergr. Space Technol. 23, 151–159. Rocscience, RocLab, 2002. Rocscience Inc., Toronto, Canada. Saiang, D., Nordlund, E., 2009a. Numerical analyses of the influence of blast-induced damaged rock around shallow tunnels in brittle rock. Rock Mech. Rock Eng. 42 (3), 421–448. Saiang, D., Nordlund, E., 2009b. Numerical analyses of the influence of blast-induced damaged rock around shallow tunnels in brittle rock. Rock Mech. Rock Eng. 42, 421–448. Saiang, D., 2010. Numerical stability analyses of the blast induced damaged zone by continuum and coupled continuum–discontinuum methods. Eng. Geol. 116 (27), 1–11. Sharan, S.K., 2003. Elastic–brittle–plastic analysis of circular openings in Hoek– Brown media. Int. J. Rock Mech. Min. Sci. 40, 817–824. Sharan, S.K., 2005. Exact and approximate solutions for displacements around circular openings in elastic–brittle–plastic Hoek–Brown rock. Int. J. Rock Mech. Min. Sci. 42, 542–549. Timoshenko, S.P., Goodier, J.N., 1982. Theory of Elasticity. McGraw-Hill, New York. Varas, F., Alonso, E., Alejano, L., Fdez-Manin, G., 2005. Study of bifurcation in problem of unloading a circular excavation in a strain softening material. Tunn. Undergr. Space Technol. 20 (4), 311–322. Zareifard, M.R., Fahimifar, A., 2012. A new solution for shallow and deep tunnels by considering the gravitational loads. ACTA Geotech. Slovenica 2 (4), 37–49. Zareifard, M.R., Fahimifar, A., 2014. Effect of seepage forces on circular openings excavated in Hoek–Brown rock mass based on a generalised effective stress principle. Eur. J. Environ. Civ. Eng. 18 (5). Zareifard, M.R., Fahimifar, A., 2015. Elastic–brittle–plastic analysis of circular deep underwater cavities in a Mohr–Coulomb rock mass considering seepage forces. Int. J. Geomech. 15 (5).
Contents lists available at ScienceDirect
Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Analytical solutions for the stresses and deformations of deep tunnels in an elastic-brittle-plastic rock mass considering the damaged zone Mohammad Reza Zareifard a,⇑, Ahmad Fahimifar b a b
Estahban Institute of Higher Education, Estahban, Iran Amirkabir University of Technology, Tehran, Iran
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 7 March 2015 Received in revised form 16 April 2016 Accepted 23 May 2016
The excavation impact (e.g. due to blasting, TBM drilling, etc.) induces an excavation damaged or disturbed zone around a tunnel. In this regard, in drill and blast method, the damage to the rock mass is more significant. In this zone, the stiffness and strength parameters of the surrounding rock mass are different. The real effect of a damage zone developed by an excavation impact around a tunnel, and its influence on the overall response of the tunnel is of interest to be quantified. In this paper, a fully analytical solution is proposed, for stresses and displacements around a tunnel, excavated in an elastic– brittle–plastic rock material compatible with linear Mohr–Coulomb criterion or a nonlinear Hoek–Brown failure criterion considering the effect of the damaged zone induced by the excavation impact. The initial stress state is assumed to be hydrostatic, and the damaged zone is assumed to have a cylindrical shape with varied parameters; thus, the problem is considered axial-symmetric. The proposed solution is used to explain the behavior of tunnels under different damage conditions. Illustrative examples are given to demonstrate the performance of the proposed method, and also to examine the effect of the damaged zone induced by the excavation impact. The results obtained by the proposed solution indicate that, the effects of the alteration of rock mass properties in the damaged zone may be considerable. Ó 2016 Elsevier Ltd. All rights reserved.
Keywords: Circular tunnel Mohr–Coulomb failure criterion Hoek–Brown failure criterion Brittle-plastic rock Analytical solution Excavation damaged zone
1. Introduction In order to analyze a tunnel, it is essential to understand the various rock mass behaviors after an excavation. The original properties of a rock or rock mass near a tunnel are changed after the excavation. The characteristics of an excavation damaged or disturbed zone (EDZ) vary with the rock mass properties, excavation method, and opening geometry. Such a disturbance can significantly influence the response of the rock mass and the overall performance of the tunnel. Therefore, investigating the influence of the EDZ around an underground excavation is of paramount importance. From this point of view, the characteristics of an EDZ have been extensively investigated for various rock engineering projects, including dam construction, tunnel construction, and waste repository projects. An EDZ can be defined as a rock zone where the rock failure and stiffness parameters have been changed due to the processes related to an excavation. Different mechanisms are related to the
⇑ Corresponding author. E-mail addresses: (A. Fahimifar).
[email protected]
(M.R.
http://dx.doi.org/10.1016/j.tust.2016.05.007 0886-7798/Ó 2016 Elsevier Ltd. All rights reserved.
Zareifard),
[email protected]
development of an EDZ. Major factors related to the development of an EDZ are (a) the excavation impact; and (b) the stress redistribution after the excavation. Based on the factor (a), both TBM and drill and blast excavation disturb the surrounding rock mass. However, in contrast to blasted tunnels, in TBM tunnels, the annular thickness of the EDZ and the rock mass alteration in the EDZ are expected to be insignificant. The extent of the damaged zone induced by excavation impact depends on the rock properties, shape of the tunnel, excavation method and its quality, etc., but it can range from few centimeters in tunnels with TBM to several decimeters and up to several meters with drill and blast (Beackblom and Martin, 1999; Martino and Chandler, 2004). For this reason, the blast-induced damaged zone (BIDZ) has to be considered in the analysis and design of tunnels; while, the TBM drilling-induced damaged rock can be ignored. However, for very high quality controlled blasting, the damage to the rock mass is negligible due to a well-designed blasting pattern and detonation sequence and accurate drilling control. In contrast, the lack of a good blast design and absence of any control on the drilling can result in the significant damage. On the other hand, based on factor (b), due to the stresses induced by the tunnel excavation, a disturbed plastic or fractured zone will develop around the tunnel. However, the elastoplastic
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187
Nomenclature D GSI p0 r Rp ri ur e1 e3 eh er
m rc
disturbance factor Geological Strength Index hydrostatic in situ stress radial distance from the center of the tunnel radius of the plastic zone radius of tunnel radial displacement major principal strain in rock mass minor principal strain in rock mass circumferential strain radial strain Poisson’s ratio of rock mass uniaxial compressive strength of intact rock
response of the tunnel is affected by the damaged zone developed by the excavation impact. It can be concluded that, the factor (b) is influenced by the factor (a). Thus, for an elastoplatic analysis of a tunnel, the effects of the BIDZ developed by the excavation impact must be taken into account. In this regard, one of the significant reasons for assessing the BIDZ around tunnels is its effect on the tunnel stability. This, therefore, implies the need for considering this zone during the tunnel design. Full consideration of the interplay that exists among construction activities, the EDZ, support characteristics, and time requires the use of numerical methods in which all factors can be considered (see, for example Cho et al., 2006; Saiang and Nordlund, 2009a, 2009b; Saiang, 2010). No closed-form solutions exist that include the full complexity of such a problem. Analytical solutions, however, if found, have the potential of providing an insight into the problem, and are very useful to identify the most important variables for a given subject, and contribute to the understanding of the excavation–rock–liner interaction problem. Analytical solutions, however, suffer from distinct limitations, because they usually require a number of assumptions and simplifications that often apply to the problems with limited practical interest. Nevertheless, the advantages of having a closed-form solution often outweigh the limitations. In this regard, no notable analytical solutions that consider the effect of the BIDZ are currently available for analyzing the elastoplastic response of tunnels. In elasto-plastic solutions, proposed by Brown and Bray (1982), Carranza-Torres and Fairhurst (1999), Sharan (2003,2005), Alonso et al. (2003), Park and Kim (2006), Park et al. (2008), Lee and Pietruszczak (2008), Fahimifar and Zareifard (2009, 2012, 2014), Zareifard and Fahimifar (2012, 2014, 2015), Alejano et al. (2010), Cheng (2012), effects of the BIDZ have not been considered, and they are presented for tunnels excavated in initially homogenous conditions. In practice, the global damage induced by blasting is taken into account by using the damage factor D introduced by Hoek et al. (2002). In this method, the blast damage factor D is applied to the entire rock mass surrounding the tunnel. This is a common modeling method which can greatly underestimate the strength and stability of the overall rock mass. In this regard, the blast damage factor D is more appropriate to be applied to the actual zone of damaged rock. In this paper, a fully analytical elastic–brittle–plastic solution for a deep circular tunnel in a Mohr–Coulomb or Hoek–Brown rock mass, considering a cylindrical and homogenous BIDZ is presented. For the models in this paper, it is assumed that the damage induced by blasting is finite in extent and is in the form of a cylindrical zone. Beyond the BIDZ, it is assumed that the rock is not damaged and; therefore, the undamaged rock property values are used.
r1 r3 rr rh
major principal stress minor principal stress radial stress circumferential stress u, C material constants for Mohr–Coulomb rock mass m, s material constants for Hoek-Brown rock mass W dilation angle Subscript ‘D’ refers to quantities corresponding to damaged rock Superscript ‘e’ refers to elastic part of strain Subscript ‘r’ denotes the values of rock mass parameters for plastic zone Subscript ‘i’ denotes the initial values of rock mass parameters Superscript ‘p’ refers to plastic part of strain
The discussion of this paper is restricted to the stable solution, so the topics concerning the instability such as bifurcation and strain localization in the plastic regime (Varas et al., 2005; Alonso et al., 2003) are not considered. In the present exact analytical solution, simple formulas are derived without having to solve complex differential equations. This is useful for acquiring an additional insight into the problem on an opening within a damaged zone; because only a minimal computational effort is needed and considerable economic benefits can be gained by using it in the preliminary stage of a tunnel design. In addition, the exact solution is useful for the verification of the numerical codes and semi-analytical solutions. Furthermore, the proposed solution can also be applied for a tunnel reinforced by means of a grouting zone with increased strength parameters.
2. Definition of the problem The problem considered is shown in Fig. 1. A circular deep tunnel of radius ri is excavated in an initially elastic rock mass characterized by Young’s modulus E, and Poisson’s ratio m. Due to a blasting impact a cylindrical blast-induced damaged zone (BIDZ) will develop around the tunnel with different behavior parameters. In this regard the Young’s modulus and the Poisson’s ratio of BIDZ are ED and mD, respectively. Axial symmetry conditions for geometry and loading can be assumed for the problem of the tunnel under a uniform initial stress p0. A uniform internal pressure ri ¼ rrðri Þ is considered to act on the periphery of the tunnel as a result of a lining installation. The stress redistribution and displacements will take place, due to excavation of the tunnel, installing the lining and the alteration of the rock mass in the BIDZ. As ri is gradually reduced, a radial displacement occurs and a plastic zone develops around the tunnel as ri becomes less than the initial yield stress. After failure, the strength of the rock suddenly drops and follows the post-failure softening behavior. In this study, both the damaged and undamaged rock masses are considered to be elastic–brittle-plastic (in special case: perfectly plastic) as shown in Fig. 2. Two different zones may forms around the tunnel: an external elastic zone and an internal plastic zone of radius Rp. In this respect, three different cases can be considered, depending on the extent of the BIDZ and the plastic zone (Fig. 1): Case (a): the radius of the plastic zone is larger than the BIDZ. Case (b): the radius of the BIDZ is greater than the plastic zone. Case (c): the radius of the BIDZ is equal to the radius of the plastic zone.
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r
BIDZ
Plastic zone
r r
r Elastic Zone
i
RD
ri
At infinity
p0
r
Rp
(a)
r
Plastic zone
r r
r
BIDZ
Elastic Zone i
ri
At infinity
r
p0
drrðrÞ rhðrÞ rrðrÞ ¼0 dr r
Rp
RD
The analyses for three cases are different. In this way, for analysis of a tunnel with an unknown plastic radius, the calculations are, firstly, carried out based on the case (a) as a choice. Then, if the calculated plastic radius becomes larger than the radius of BIDZ; the chosen case is correct, and the results are acceptable. Otherwise, the case (b) should be taken into account. Now, if the calculated plastic radius is smaller than the radius of the BIDZ, the results are acceptable. Otherwise, the case (c) must be considered. In the proposed solution, the effect of the gravitational loadings in the plastic zone is ignored. However, in highly fractured and damaged rock masses, as shown by Carranza-Torres and Fairhurst (1997) and Zareifard and Fahimifar (2012), the weight of the plastic zone may affect the response of the rock mass. Considering the axial-symmetry condition of the problem in Fig. 1 (three cases), the resulting stress state at a distance r is defined by the radial stress rr(r) and circumferential stress rh(r) which are the minor and major principal stresses, r3(r) and r1(r), respectively. The resulting displacement is defined by the radial displacement ur(r). It should be noted that for such long tunnel problem, the out-of-plane stress rz(r), can be taken as the intermediate principal stress r2(r). For the considering axial-symmetric problem, the equilibrium equation relating the radial stress rr(r) and the circumferential stress rh(r) at the radial distance r is (Timoshenko and Goodier, 1982):
The radial strain er(r) and circumferential strain eh(r) at radius r for axial-symmetric condition can be stated in terms of the radial displacement ur(r), as follows (Timoshenko and Goodier, 1982):
(b)
eh ¼ r
r r
r
Plastic zone
Elastic Zone
BIDZ i
ri
At infinity
r
Rp
ð1Þ
p0
RD
(c) Fig. 1. The circular deep tunnel subjected to a hydrostatic stress field with a cylindrical BIDZ: (a) Rp > RD, (b) Rp < RD, and (c) Rp = RD.
ur ; r
er ¼
dur dr
ð2Þ
For both rock masses (the BIDZ and undamaged zone), the induced elastic strains eer and eeh in terms of the final stresses rr and rh with consideration of the initial hydrostatic stress p0, can be written as (Timoshenko and Goodier, 1982):
1þm ½ð1 mÞðrr p0 Þ mðrh p0 Þ E 1þm ¼ ½ð1 mÞðrh p0 Þ þ mðrr p0 Þ E
eer ¼
ð3Þ
eeh
ð4Þ
where E and m are Young’s modulus and Poisson’s ratio of the rock mass, respectively. In this regard, rock mass parameters for the BIDZ are denoted by a subscript ‘D’; namely, the elastic parameters for the BIDZ are ED and mD.
Fig. 2. Behavior model used in this study for both the damaged and undamaged rock masses: (a) Stress-strain relationship. (b) Flow rule.
M.R. Zareifard, A. Fahimifar / Tunnelling and Underground Space Technology 58 (2016) 186–196
For the plastic regime, two most commonly used failure criteria are considered: the nonlinear Hoek-Brown failure criterion (Hoek et al., 2002):
r1 r3 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mr3 rc þ sr2c
ð5-1Þ
and the linear Mohr–Coulomb failure criterion,
r1 ¼ A þ Br3
ð5-2Þ
cos u u ; r1 and r3 are the major and minor where A ¼ 2C ; B ¼ 1þsin 1sin u 1sin u
principal stresses at failure, respectively; rc is the uniaxial compressive strength of the intact rock material; m and s are material constants for the Hoek-Brown failure criterion; and C and u are material constants for the Mohr–Coulomb failure criterion (C and u are the cohesion and the friction angle of the rock, respectively). For the present axial-symmetry solution, principal stresses, r1 and r3 are equal to the radial and circumferential stresses rr and rh, respectively. Eq. (5) may thus be rewritten as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rh rr ¼ mrr rc þ sr2c rh ¼ A þ Brr
ð6-1Þ ð6-2Þ
rh rr r h ¼ Ai þ B i r r
ð7-1Þ ð7-2Þ
ui i cos ur ; and Ci, ui and mi, si are material where Ai ¼ 2C ; Bi ¼ 1þsin 1sin u 1sin u i
i
constants for the initial rock mass. And for the plastic zone r i 6 r 6 Rp (Fig. 1), Eq. (6) is modified as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rh rr ¼ mr rr rc þ sr r2c rh ¼ Ar þ Br rr
ð8-1Þ ð8-2Þ
ur r cos ur ; and Cr, ur and mr, sr are material where Ar ¼ 2C ; Br ¼ 1þsin 1sin u 1sin u r
r
constants for the failed rock mass. The failure parameters Ci, ui and mi, si and the elastic parameter E vary with rock type and rock mass quality as measured by is the Geological Strength Index of the rock mass (GSI) (Hoek et al., 2002). In the broken or plastic zone, the reduced strength parameters Cr, ur and mr, sr can be obtained using the residual GSI according to a method proposed by Alejano et al. (2009). It should be noted that the damaged rock mass has different failure parameters, here denoted by subscript ‘D’ (namely, rcD, CiD, uiD, miD, siD, CrD,urD, mrD, srD). These parameters can be attained considering a suitable damage factor (D) (Hoek et al., 2002). In this regard, the blast damage factor was first introduced in the 2002 version of the Hoek-Brown criterion (Hoek et al., 2002) and it is used for the estimation of the strength and deformability parameters. In the plastic zone, the total radial and circumferential strains, er and eh, can be divided into the elastic and plastic parts:
er ¼ eer þ epr ; eh ¼ eeh þ eph
ð9Þ
By assuming that the elastic strains are relatively small as compared to the plastic strains and that a non-associated flow rule is valid, the plastic parts of radial and circumferential strains may be related as (Sharan, 2003; Park and Kim, 2006):
epr þ K W eph ¼ 0
Most of the researchers who have studied dilatancy in rock and rock masses have acknowledged the need to use non-associative flow rules. Given that the performance of significant tests on rock masses is a complex matter, dilatancy studies should be based on field experiments and back-analysis (Alejano and Alonso, 2005). In this regard, the analytical solutions like the present closedform solution can be beneficial. Based on wide practical rock engineering experience and probably inspired by the idea of providing the rock mechanics community with parameters for feeding numerical models, Hoek and Brown (1997) recommended the use of constant dilatancy angle values based on rock mass quality; they proposed, thus, values of W = u/4 for excellent- and goodquality rock masses, W = u/8 for average-quality rock masses, and W = 0 for poor-quality rock masses. From Eqs. (2) and (10), the displacement compatibility equation can be expressed as (Park and Kim, 2006):
dur ur þ K W ¼ f ðrÞ dr r
ð11Þ
where
f ðrÞ ¼ B1 þ B2 rrðrÞ þ B3 rhðrÞ
ð12Þ
B1 ¼ ð1þmÞð12EmÞð1þK W Þ p0
For the initial rock mass, Eq. (6) can be written as:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ mi rr rc þ si r2c
189
ð10Þ
where KW = (1 + sin W)/(1 – sin W), and W is the dilation angle of the rock mass (in the damaged zone the dilation angle is shown by WD).
B2 ¼ ð1þmÞð1E mK W mÞ
B3 ¼ ð1þmÞðK WEmK W mÞ It should be noted that, the damaged rock mass has altered parameters. Thus for the damaged zone the following coefficients are used:
B1D ¼ ð1þmD Þð12EDmD Þð1þK WD Þ p0 B2D ¼ B3D ¼
ð1þmD Þð1mD K W
m Þ D D
ED ð1þmD ÞðK WD mD K W
m Þ D D
ED
3. Analysis of the elastic zone As shown in Fig. 1, outside the plastic zone the rock mass remains elastic. The solutions are different for the cases of (a), (b) and (c). 3.1. Cases (a) and (c) An external elastic zone of the inner radius Rp with unaltered parameters exists outside the plastic zone. The stresses and radial displacements in this zone, can be derived based on the thickwalled cylinder theory in Elasticity (Timoshenko and Goodier, 1982):
R2p r2 R2p rhðrÞ ¼ p0 þ ðp0 rrðRp Þ Þ 2 r R2p 1þm ðp0 rrðRp Þ Þ urðrÞ ¼ E r
rrðrÞ ¼ p0 ðp0 rrðRp Þ Þ
ð13Þ ð14Þ ð15Þ
3.2. Case (b) The elastic zone consists of an external undamaged zone with unaltered parameters and an internal BIDZ. As shown in Fig. 1b, the undamaged and damaged elastic zones interact with each other at the radius RD with a radial stress rrðRD Þ .
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The stresses and displacements in the undamaged zone and BIDZ are obtained using the thick-walled cylinder in Elasticity:
R2D r2 R2D rhðrÞ ¼ p0 þ p0 rrðRD Þ 2 r R2D 1 þ m urðrÞ ¼ p0 rrðRD Þ E r
rrðrÞ ¼ p0 p0 rrðRD Þ
ð16Þ ð17Þ ð18Þ
for the undamaged zone, and
rrðrÞ ¼ rrðRD Þ rrðRp Þ
rhðrÞ ¼ rrðRD Þ rrðRp Þ
R2p
R2p
R2 1 2D r
R2D R2p
R2D r2
R2D
R2p
1þ
! þ rrðRD Þ
ð19Þ
þ rrðRD Þ
ð20Þ
!
R 1 þ mD R2 ¼ rrðRD Þ rrðRp Þ 2 p 2 ð1 2mD Þ þ D ED r RD Rp 2
urðrÞ
!
1 þ mD þ ðrrðRD Þ p0 Þð1 2mD Þr ED
a1 ¼
m
2 DÞ
ðED ð1 þ mÞðR2D R2p Þ Eð1 þ mD Þðð2mD 1ÞR2D R2p ÞÞ
ð26-1Þ
for Mohr–Coulomb failure criterion and,
rrðrÞ ¼ ri þ MD log
r 2 r þ N D log ri ri
ð26-2Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! MD ¼ mrD ri rcD þ sr r2cD . ND ¼ mrD4rcD Then, solving the differential equation for undamaged zone leads to the following equation: for Hoek-Brown failure criterion, where
rrðrÞ ¼
!, 1BrD RD ð1 Br Þ Ar þ ð1 Br ÞrrðRD Þ þ Ar r
ð27-1Þ
for Mohr–Coulomb failure criterion and,
r r 2 þ Nlog RD RD
ð27-2Þ
ð21Þ
ð22Þ
2R2p Eð1
r 1BrD i ðð1 BrD Þri þ ArD Þ ð1 BrD Þ r
rrðrÞ ¼ rrðRD Þ þ M log
for the BIDZ. The boundary pressure rrðRD Þ between these elastic zones is obtained by using the equilibrium and compatibility conditions at the boundary between them, i.e. at radius RD.
rrðRD Þ ¼ a1 ðrrðRD Þ p0 Þ þ p0
rrðrÞ ¼ ArD þ
ð23Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M ¼ mr rrðRD Þ rc þ sr r2c . mr rc N¼ 4 In Eqs. (27), the radial stress rrðRD Þ is obtained using Eqs. (26). The corresponding value of the circumferential stress rh,is calculated using the failure criterion (i.e. Eqs. (8)). The radius of the plastic zone Rp can be obtained by considering the continuity of the radial stress at the elastic–plastic interface. Consequently, the radial stress obtained at the plastic radius from Eqs. (27) must be equal to the corresponding value in Eq. (25); because, the radial stress must be continues over the boundary. Thus, the following equation for the radius of the plastic zone, Rp is derived: for Hoek-Brown failure criterion, where
Rp ¼ RD
4. Analysis of the plastic zone
0
4.1.1. Case (a) The plastic zone consists of an external undamaged zone with unaltered parameters and an internal damaged zone (Fig. 1a). As shown in Fig. 1a, the undamaged and damaged zones interact with each other at the radius RD with a radial stress rrðRD Þ . The elastic and plastic zones interact with each other at the plastic radius Rp, where the radial boundary pressure rrðRp Þ is applied. Utilizing Eq. (14), the circumferential stress rhðRp Þ at the radius Rp, is obtained as:
rhðRp Þ ¼ 2p0 rrðRp Þ
ð24Þ
rhðRp Þ and rrðRp Þ must satisfy the failure criterion for the undamaged rock; therefore, substituting these stresses into Eq. (7), and solving the derived equation, gives the boundary radial stress,
rrðRp Þ ¼ ð2p0 Ai Þ=ðBi þ 1Þ
ð25-1Þ
for Mohr–Coulomb failure criterion and,
rrðRp Þ
ð28-1Þ
for Mohr–Coulomb failure criterion and,
4.1. Stresses
1 ¼ p0 þ rc mi 8
1 Ar rrðRp Þ ð1 Br Þ 1Br Ar rrðRD Þ ð1 Br Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 16mi p0 þ m2i þ 16si
rc
ð25-2Þ
for Hoek-Brown failure criterion. After obtaining the value of rrðRp Þ from Eq. (25), the corresponding value of rhðRp Þ is obtained from Eq. (24) or Eq. (7). Then, the radial displacement urðRp Þ is obtained using Eq. (15). By substituting Eq. (8) into Eq. (1) and solving the derived differential equation by using the boundary condition rrðri Þ ¼ ri at r = ri, the following closed-form equation for the radial stress rr(r) in the damaged area of the plastic zone ðr 6 RD Þ is obtained:
Rp ¼ RD exp @
M þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 M 2 4NðrrðRD Þ rrðRp Þ Þ A 2N
ð28-2Þ
for Hoek-Brown failure criterion 4.1.2. Case (b) The elastic zone and the damaged plastic zone interact with each other at the plastic radius Rp, where the radial boundary pressure rrðRp Þ is applied. Utilizing Eq. (20) by considering Eq. (22), the circumferential stress rhðRp Þ at the radius Rp, is obtained as:
rhðRp Þ ¼ ð1 þ aÞp0 arrðRp Þ
ð29Þ
a ¼ a2 a1 a1 a2 where a1 is obtained from Eq. (23) and,
a2 ¼
R2D þ R2p R2D R2p
rhðRp Þ and rrðRp Þ must satisfy the failure criterion for the damaged rock; therefore, substituting these stresses into Eq. (7), and solving the derived equation, gives the boundary radial stress,
rrðRp Þ ¼
ða þ 1Þp0 AiD BiD þ a
ð30-1Þ
for Mohr–Coulomb failure criterion and, 0
rrðRp Þ ¼ p0 þ
1
r @mi 2 cD
2ða þ 1Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 4ða þ 1Þ2 miD p0 þ m2iD þ 4ða þ 1Þ2 siD A
rcD
ð30-2Þ for Hoek-Brown failure criterion
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After obtaining the value of rrðRp Þ from Eq. (30), the corresponding value of rhðRp Þ is obtained from Eqs. (29) or (7). Then, the radial displacement urðRp Þ is obtained using Eq. (21). Solving Eq. (1) for the failed rock mass (using Eq. (8)) applying the boundary condition rrðri Þ ¼ ri at r = ri, the radial stress rr(r) in the damaged plastic zone is obtained from Eq. (26). The corresponding value of the circumferential stress rh, is calculated using the failure criterion (i.e. Eqs. (8)). The radial stress obtained at the plastic radius from Eqs. (26) must be equal to the corresponding value in Eq. (30); because, the radial stress must continue over the boundary. Thus, the following equation for the radius of the plastic zone, Rp is derived,
1 ArD rrðRp Þ ð1 BrD Þ 1BrD Rp ¼ r i ArD ri ð1 BrD Þ
On the other hand, solving the differential Eq. (11) for the damaged area ðr 6 RD Þ gives:
ur ¼
1 rK WD
ðFðrÞ FðRD ÞÞ þ urðRD Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 MD þ M 2D 4ND ðri rrðRp Þ Þ A Rp ¼ r i exp @ 2ND
rD ÞðB2D þB3D BrD Þ 3D BrD Þ B1D þ B3D ArD ArD ðBðB2DrDþB þ ðri ðBrDðK1ÞþA r ðK WD þ1Þ 1Þ W þBrD ÞðBrD 1Þ
FðrÞ ¼
D
ðK WD þ 1Þ
and for Hoek-Brown failure criterion:
0 B @
FðrÞ ¼ r ðK WD þ1Þ
1 2 ðK WD þ 1Þ2 B4D þ B5D log rri þ B6D log rri C A r ðK WD þ 1Þ 2B6D log r þ B5D þ 2B6D i
B4D ¼ B1D þ B2D ri þ B3D ðri þ MD Þ ð31-2Þ
for Hoek-Brown failure criterion In case (b), since, the radial stress at the plastic radius obtained from Eq. (30) is a function of the plastic radius, Rp, the solution is carried out by iteration. A trial can be attempted with an assumed value of Rp, then, the values of the radial stress and the plastic radius are calculated from Eqs. (30) and (31), alternately, to achieve a certain convergence.
ð33Þ
where for Mohr–Coulomb failure criterion,
ð31-1Þ
for Mohr–Coulomb failure criterion and,
K W D RD r
ðK W þ1Þ3 D
B5D ¼ B2D MD þ B3D ð2ND þ M D Þ B6D ¼ ðB2D þ B3D ÞND 4.2.2. Cases (b) and (c) Solving the differential Eq. (11) for the damaged plastic zone ðr 6 Rp Þ gives:
ur ¼
1 rK WD
ðHðrÞ HðRp ÞÞ þ urðRp Þ
K W D Rp r
where for Mohr–Coulomb failure criterion, 4.1.3. Case (c) In this case, the plastic radius is known, RD = Rp. The elastic zone and the damaged plastic zone interact with each other at the plastic radius RD = Rp, where the radial boundary pressure rrðRp Þ is applied. Solving Eq. (1) for the failed rock mass (considering Eq. (8)) using the boundary condition rrðri Þ ¼ ri at r = ri, the radial stress rr(r) in the damaged plastic zone is obtained from Eq. (26). The corresponding value of the circumferential stress rh, is calculated using the failure criterion (i.e. Eqs. (8)). For the plastic radius, obtaining the value of rrðRp Þ from Eq. (26), the corresponding value of rhðRp Þ is obtained from Eq. (8). Then, the radial displacement urðRp Þ is obtained using Eq. (15). 4.2. Displacements 4.2.1. Case (a) Solving the differential Eq. (11) by using the boundary radial displacement at the plastic radius urðRp Þ , the following closedform equation for the radial displacement ur(r) in the undamaged area of the plastic zone ðRD 6 r 6 Rp Þ is obtained:
ur ¼
1 rKW
ðGðrÞ GðRp ÞÞ þ urðRp Þ
K W Rp r
where for Mohr–Coulomb failure criterion:
GðrÞ ¼
ðrrðR Þ ðBr 1ÞþAr ÞðB2 þB3 Br Þ 3 Br Þ D B1 þ B3 Ar Ar ðBðB2rþB þ rðK W þ1Þ ðK W þBr ÞðBr 1Þ 1Þ ðK W þ 1Þ
and for Hoek-Brown failure criterion:
1 2 ðK W þ 1Þ2 B4 þ B5 log RrD þ B6 log RrD @ A ðK W þ 1Þ 2B6 log RrD þ B5 þ 2B6 0
GðrÞ ¼ rðK W þ1Þ
ðK W þ1Þ3
B4 ¼ B1 þ B2 rrðRD Þ þ B3 ðrrðRD Þ þ MÞ B5 ¼ B2 M þ B3 ð2N þ MÞ B6 ¼ ðB2 þ B3 ÞN
ð32Þ
HðrÞ ¼
rD ÞðB2D þB3D BrD Þ 3D BrD Þ B1D þ B3D ArD ArD ðBðB2DrDþB þ ðri ðBrDðK1ÞþA rðK WD þ1Þ 1Þ W þBrD ÞðBrD 1Þ D
ðK WD þ 1Þ ð40-1Þ
and for Hoek-Brown failure criterion:
0 B @
HðrÞ ¼ r ðK WD þ1Þ
1 2 ðK WD þ 1Þ2 B4D þ B5D log rr þ B6D log rr i i C A r ðK WD þ 1Þ 2B6D log ri þ B5D þ 2B6D
B4D ¼ B1D þ B2D ri þ B3D ðri þ MD Þ
ðK W þ1Þ3 D
B5D ¼ B2D MD þ B3D ð2ND þ M D Þ B6D ¼ ðB2D þ B3D ÞND 5. Practical applications In order to investigate the effect of the BIDZ on the displacement and stresses, the results obtained from the solution for cases of hard and soft rocks, are compared using different disturbance conditions. Here, properties of rocks and data for underground openings are taken from the published sources (Carranza-Torres and Fairhurst, 1999; Alejano et al., 2009). 5.1. A tunnel in a soft elastic–perfectly plastic rock mass Here, the proposed solution is used for parametric study to observe the effects of the BIDZ on the overall response of a tunnel in a soft elastic–perfectly plastic rock mass. The initial rock mass parameters used in the failure criteria can be estimated by determining the unconfined compressive strength of intact rock samples rc, the Geological Strength Index (GSI), the rock parameter mintact (which is a Hoek-Brown material constant for intact samples), and the Disturbance factor (D). The Rocscience program RocLab is capable of estimating rc and mintact by either fitting laboratory data or by using built-in charts of typical parameter
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ranges. The GSI (Hoek et al., 2002) is a parameter that is used to help link the mi and si parameters to field observations of the rock mass. The disturbance factor D, which varies from zero (e.g., an excellent quality controlled blasting or excavation) to one (e.g., a significant disturbance due to heavy production blasting), is used to capture damage due to blasting or disturbance. One of the data sets appearing in (Carranza-Torres and Fairhurst, 1999; Sharan, 2003) is taken as this input data. An intact rock core, sampled from the rock mass and tested in the laboratory, indicates the following (intact rock) properties:
rc ¼ 30 MPa; mintact ¼ 10; v ¼ 0:25 The rock mass has a GSI value of 50. All of these parameters are used to determine mi, si, ui and Ci respectively. In this way, considering the elastic–perfectly plastic behavior, the ‘scaling’ failure parameters in Mohr-Coulomb and Hoek-Brown failure criteria for the undamaged rock mass can be obtained using RocLab (Rocscience, 2002) considering D = 0. Hoek-Brown parameters:
mi ¼ mr ¼ 1:7;
si ¼ sr ¼ 0:0039;
rc ¼ 30 MPa:
miD ¼ mrD ¼ 0:8;
siD ¼ srD ¼ 0:001;
rcD ¼ 30 MPa:
Mohr-Coulomb parameters:
uiD ¼ urD ¼ 20 ; C iD ¼ C rD ¼ 1:5 MPa: These values are obtained using the damage factor of D = 0.6 and similar GSI and intact rock material properties with undamaged rock. In addition, Poisson’s ratio assumed to be unaltered for the damaged zone, in contrast to the initial rock mass, i.e. vD = 0.25. A supported circular tunnel of radius ri = 5 m, excavated in a rock mass having the mechanical properties described above, is considered. The far-field stress is assumed to be p0 = 30 MPa and the tunnel is supported by an internal pressure of ri ¼ 5 MPa. In order to investigate the effect of the thickness of the BIDZ on the stresses and displacements in the rock mass, the results are compared for different radial extents of the BIDZ. The dimensionless radial displacement, urir ½% and the dimensionless stresses rrh and rrr , in the rock mass are plotted in i
Mohr-Coulomb parameters:
ui ¼ ur ¼ 26 ; C i ¼ C r ¼ 2:1 MPa: On the other hand, for the rock mass considered in this example (GSI = 50 and rc ¼ 30 MPa), RocLab indicates a value of E ¼ 5500 MPa. Both damaged and undamaged rock masses obey a nonassociated flow rule with no plastic dilation, i.e. W = WD = 0. It should be notified that, in the proposed models, it is assumed that, the disturbance is in the form of the blast induced-damaged zone around the tunnel boundary. Here, for estimating the rock mass parameters in the damaged zone the notion of disturbance factors, D, (Hoek et al., 2002) is used. It is assumed that the damaged rock mass is characterized by a D value of 0.6; thus, using RocLab (Rocscience, 2002), the rock mass parameters for the blast-induced damaged zone are obtained as: Young’s modulus:
ED ¼ 3800 MPa
Fig. 3. Dimensionless radial displacement,
Hoek-Brown parameters:
ur ri
i
Figs. 3–6 for cases RD = 5 m, RD = 7 m, RD = 10.5 m and RD = 15 m. While the high values chosen for the extent of the BIDZ may not be usually encountered in practice, choosing these high values, e.g. RD = 15 in this study is to investigate the maximum possible effect of BIDZ thickness on the predicted displacements and stresses, and to illustrate cases (b) and (c) (see Fig. 1(b and c)). For RD = 5m the rock mass remains totally undamaged. On the other hand, for RD = 7 m, RD = 10.5 m and RD = 15 m, the results show that cases (a), (c) and (b) should be used, respectively. It is observed that, the radius of the damaged zone and degree of disturbance in this zone are the most important parameters. However, measuring these parameters is difficult, especially in the early design of tunnels. Hence, incorporating the damaged zone in the early design stage of tunnels is often neglected. One of the main advantages of the proposed solution is that, in contrast to standard FDM or FEM-based codes, it permits performing parametric studies more conveniently. Parametric studies are usually very recommendable in such cases where a certain deal of uncertainty is always expected. This kind of analysis is very helpful for
½% (for soft rock): (a) Hoek-Brown rock mass and (b) Mohr-Coulomb rock mass.
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193
Fig. 4. Dimensionless circumferential stresses rrhi (for soft rock): (a) Hoek-Brown rock mass and (b) Mohr-Coulomb rock mass.
Fig. 5. Dimensionless radial stresses rrri (for soft rock): (a) Hoek-Brown rock mass and (b) Mohr-Coulomb rock mass.
controlling uncertainties encountered in the quantification of the damaged zone. In this way, Figs. 6 and 7 illustrate the effect of the disturbance factor (D) and the radius of the damaged zone on the radius of the plastic zone and the tunnel convergence, respectively. The corners observed in the diagrams are because of the fact that at these points the radii of the damaged zone become equal to the radii of the plastic zone (i.e. transfer to another case). The results show that, if a wide damaged zone with high reduced parameters develops around the tunnel, its influence is significant on the stresses and displacements around the tunnel
(i.e. for RD > 1.5ri and D P 0.5). In Figs. 3–7, the results for both Hoek-Brown and Mohr-Coulomb failure criteria are plotted. The difference in the results of those failure criteria seems insignificant, especially when RD is low.
5.2. A tunnel in a hard elastic–brittle–plastic rock mass Now, the effects of the BIDZ on the response of a tunnel in a hard Hoek–Brown elastic–brittle–plastic rock mass are investigated.
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Fig. 6. Influence of disturbance factor (D) and radius of the damaged zone (RD) on radius of the plastic zone Rp (for soft rock): (a) Hoek-Brown rock mass and (b) MohrCoulomb rock mass.
Fig. 7. Influence of disturbance factor (D) and the radius of the damaged zone (RD) on tunnel convergence urðri Þ (for soft rock): (a) Hoek-Brown rock mass and (b) Mohr-Coulomb rock mass.
One of the data sets appearing in (Alejano et al., 2009) is taken as this input data. An unsupported circular tunnel of radius ri = 7 m is excavated in a rock mass (GSI = 64.9) composed of conglomerate, sandstone and mudstone to a depth of 1000 m, where an in situ hydrostatic stress p0 = 27 MPa is applied. Laboratory testing indicated values for rc = 162 MPa, mintact = 19, v = 0.25. These parameters are used to determine Young’s modulus and the initial Hoek-Brown parameters for the undamaged rock mass, E, mi, si, using RocLab (Rocscience, 2002) considering D = 0:
mi ¼ 5:4;
si ¼ 0:02;
E ¼ 24 GPa:
For estimating these parameters in the damaged zone, it is assumed that the disturbed rock mass is characterized by a D value of 0.6:
miD ¼ 3:2;
siD ¼ 0:0076;
ED ¼ 16 GPa:
It is assumed that, in the damaged zone the parameters rc and v remain unchanged. According to a method proposed by Alejano et al. (2009), the residual GSI is 27.8, and the corresponding residual
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Fig. 8. Dimensionless radial displacement,
ur ri
½% for the tunnel in hard rock.
Fig. 10. Dimensionless radial stresses rrr for the tunnel in hard rock. i
Fig. 9. Dimensionless circumferential stresses
rh ri
for the tunnel in hard rock.
Hoek-Brown strength parameters are mr = 1.4, mrD = 0.5, sr = 0.0003, srD = 0.00004. Both damaged and undamaged rock masses obey a nonassociated flow rule with dilatancy angles W ¼ WD ¼ 15 . In order to investigate the effect of the thickness of the BIDZ on the stresses and displacements in the rock mass, the results are compared for different radial extents of the BIDZ. The dimensionless radial displacement, urir ½% and the dimensionless circumferential stress rrh , in the rock mass are plotted in i
Fig. 11. Influence of disturbance factor (D) and radius of the damaged zone (RD) on radius of the plastic zone Rp for the tunnel in hard rock.
Figs. 8–10 for cases RD = 7, RD = 8, RD = 9.5 and RD = 12. For RD = 7 m the rock mass remains totally undamaged. On the other hand, for RD = 8, RD = 9.5 and RD = 12, the results show that case (a), (c) and (b) must be used, respectively. Figs. 11 and 12 illustrates the effect of the disturbance factor (D) and the radius of the damaged zone on the radius of the plastic zone and the tunnel convergence, respectively. As observed, in the hard rock similar results are obtained, namely for RD > 1.5ri and D P 0.5, the effect of the damaged zone is not negligible.
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Fig. 12. Influence of disturbance factor (D) and the radius of the damaged zone (RD) on tunnel convergence urðri Þ for the tunnel in hard rock.
6. Conclusions In this work, simple closed-form analytical solutions of the displacements and stresses for the elastic–brittle–plastic analysis of a circular tunnel excavated in Hoek–Brown or Mohr–Coulomb rock mass were introduced in a theoretically consistent way. In the proposed method, the effect of a homogeneous and isotropic damaged zone induced by the excavation impact (mainly, due to blasting), around the tunnel was taken into account. Since the radius of the plastic zone is unknown initially; three different cases were introduced depending on the radii of the blastinduced damaged and the plastic zones. The solutions for these cases were derived, and the results were compared for tunnels in soft and hard rock masses. The results, obtained by using the proposed solution, indicate that a blast-induced damaged zone has a significant effect on the stresses and displacements in the rock mass, when the disturbance and the radius of the damaged zone are relatively high. References Alejano, L.R., Alonso, E., 2005. Considerations of the dilatancy angle in rocks and rock masses. Int. J. Rock Mech. Min. Sci. 42 (4), 481–507. Alejano, L.R., Alonso, E., Dono, A.R., Feranndez-Man, G., 2009. Ground reaction curves for tunnels excavated in different quality rock masses showing several types of post-failure behaviour. Tunn. Undergr. Space Technol. 24, 689–705. Alejano, L.R., Alonso, E., Dono, A.R., Feranndez-Man, 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.
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