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Analytical solution for a circular opening in a rock mass obeying a three-stage stress–strain curve
International Journal of Rock Mechanics & Mining Sciences 86 (2016) 16–22
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Technical Note
Analytical solution for a circular opening in a rock mass obeying a three-stage stress–strain curve Qiang Zhang n, Binsong Jiang, Huajian Lv 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
art ic l e i nf o Article history: Received 17 September 2015 Received in revised form 14 March 2016 Accepted 22 March 2016 Keywords: Brittle rock mass Circular opening Yield criterion Analytical solution
1. Introduction The displacements and stresses of the surrounding rock generally influence the stability of the underground engineering. A great number of research studies have attempted to predict the evolution law by examining different methods. Among these studies, an analytical method of elasto-plastic theory was the most widely used for the axisymmetric circular openings with the linear Mohr–Coulomb (M–C) criterion, as well as the nonlinear Hoek– Brown (H–B) criterion.1 Brown et al.2 Wang,3 and Carranza–Torres and Fairhurst4 proposed analytical and numerical solutions based on elastic-perfectly-plastic models, respectively. At a later point in time, Sharan5 pointed out the defects in the woprk of 2,3, and then derived the closed-form solutions of circular openings in H–B rock masses5. Park et al.6 studied the displacements of the surrounding rock with four types of elastic strain distribution forms in the plastic region. However, only the solutions based on the generalized Hooke's law were found to be correct. Similarly, utilizing the assumption of the same elastic strain distribution in thick cylinders, Sharan derived analytical solutions for circular openings using a generalized H–B criterion.7 Furthermore, Chen and Tonon8 studied the analytical solutions of circular openings in generalized H–B rock masses. However, it was determined that the solution could n
Corresponding author. E-mail address: [email protected] (Q. Zhang).
http://dx.doi.org/10.1016/j.ijrmms.2016.03.013 1365-1609/& 2016 Elsevier Ltd. All rights reserved.
be solved with a simple numerical procedure. Wang and Yin9 derived the analytical solutions for spherical cavities excavated in elasto-brittle-plastic rock masses. Along with the analytical method, a numerical method was used for the more complicated engineering. Hajiabdolmajid et al.10 proposed a CWFS model for the failure characteristics of brittle rock masses. This model was utilized to simulate the well-documented Mine-by Experiment, which demonstrated good accordance with the in situ conditions. Also, the strain-softening analysis of the circular openings was studied for the soft rock masses using both analytical and numerical methods.11,12 It is known that the above results are mainly dependent on the constitutive model. All of the former research studies regarding brittle rock masses were almost entirely based on elasto-plastic (EPM) and elasto-brittle-plastic (EBM) models. Furthermore, the majority of the laboratory tests determined that the brittle failure characteristics usually occur when the confining pressure is low. However, with the increase in confining pressure, brittle rock masses first of all usually experience a stable perfect-plastic stage before a brittle failure occurs. Moreover, the perfect plastic (even hardening) behavior would tend to appear once the confining pressure becomes very high. This is referred to as the brittle-ductile transformation process.13 In regards to shallow underground engineering, an elasto–brittle–plastic model has the ability to effectively reflect the mechanical behavior of brittle rock masses. However, it cannot be overlooked that with increasing depth, the plastic bearing capacity is gradually enhanced, especially in
Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 86 (2016) 16–22
17
Fig. 1. Triaxial tests of brittle rock mass with various confining pressures.
deep underground engineering. Moreover, the deteriorated elastic parameters should also be considered in plastic rock masses.14–17 Therefore, this current study mainly deals with a constitutive model of brittle rock mass with consideration given to the plastic bearing capacity and deterioration of the elastic parameters. Moreover, the analytical solutions for circular openings, which were based on the EBPM proposed model, were also derived.
2. Proposed constitutive model 2.1. Triaxial tests Fig. 1 illustrates the completed strain-stress curves of marble in Jinping, which had a depth of 2550 m, and was under different confining pressures.18 It can be seen that the rock first experienced an elastic stage, and then the ideal plastic behavior occurred when it satisfied the yield function. In other words, the marble possessed a certain ideal peak plastic bearing capacity, and the plastic bearing capacity was enhanced with the increase of the confining pressure. As the deformation increased, the brittle failure occurred once the plastic strain reached a certain value. Then, the residual macro-plastic stage began. Therefore, the mechanical behavior for brittle rock masses can be represented by three stages as follows: elastic stage, peak plastic stage, and residual plastic stage, as shown in Fig. 2. Additionally, it was determined that the elastic modulus and Poisson's ratio of the peak plastic stage may be different from those of the residual stage due to cracking. It was therefore obvious that the proposed EPBM combined the EPM and EBM with the brittle failure conditions. 2.2. Mechanical parameters A great deal of previous research has determined that the Poisson's ratio has only a minimal influence on the deformation of rock masses. Therefore, it is assumed as a constant value in this current study. The calculated elastic moduli of the rock masses were 37.13 GPa, 46.33 GPa, 47.55 GPa, and 52.20 GPa for the confining pressures of 5 Mpa, 10 MPa, 20 MPa, and 40 MPa, respectively. Also, a mean value of 45.80 GPa was employed. Similarly, the Poisson's ratio was taken as 0.263. Meanwhile, the dilation angle could be easily calculated by using the maximum and minimum plastic strain, and the average values of 34.46° and 12.42° were taken for the peak plastic and residual plastic rock masses, respectively. It should be noted that the initial stage of the axial strain–stress curve with a confining pressure of 10 MPa showed a significantly nonlinear compression effect induced by the rock samples’ unevenness. Therefore, an axial compression
Fig. 2. Material behavior model for EPBM. (a) Stress–strain curve and (b) strain curve.
strain of 0.0015 was considered for this study. The peak and residual strengths determine the state of the rock mass, and it is the foundation for calculating the plastic strain. The peak and residual strengths with linear Mohr–Coulomb criterion using the simplified triaxial tests curves is shown in Fig. 3. The peak cohesion cp and friction angle φp were 35.53 MPa and 35.60°, respectively. Meanwhile, the residual cohesion cr and friction angle φr were 7.38 MPa and 42.03°, respectively. The marble showed a brittle-ductile character with the increases in the confining pressure. The residual strength equaled the peak strength when the confining pressure s3 ¼61.27 MPa. When s3 461.27 MPa, the proposed model showed a strain-hardening property. In order to ensure the correctness of the strain-hardening condition, high confining pressure tests with s3 4 61.27 MPa are recommended. 2.3. Critical failure condition As stated previously, an irreversible shear plastic strain was employed to describe the critical brittle failure condition. When the brittle failure occurred, the shear plastic strain reached its ultimate value γ p*. Theoretically speaking, when the confining pressure was large enough, the mechanical behavior of the marble
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Fig. 3. Fitting curves of peak and residual strength with Mohr–Coulomb criterion.
turned from brittle failure, to ductile failure. Therefore, there was a maximum limit value. In this way, γ p* can be formulated as follows:
γ p * = f ( σ¯3) = γ0p + Me N ⋅ σ¯ 3
(1)
where γ is the ultimate shear plastic strain of the peak plastic rock mass; σ3is the minimum principal pressure; and γ0p , M, and N are the fitting parameters. The evolution function for γp* could be easily obtained through triaxial compression tests. The calculated ultimate shear plastic strain, as well as the fitting curve, could be represented by γ p * = 0.061 − 0.061e−0.026 ⋅ σ¯ 3, as shown in Fig. 4.
Fig. 5. Calculated model.
p*
3. Analytical solution of circular opening 3.1. Theoretical equations Fig. 5 illustrates a circular opening of the radius R0 excavated in a homogeneous, isotropic rock mass subject to a hydrostatic pressure p. The inner surface of the opening was subject to a uniform pressure, which was supplied by the elastic lining structure. When the opening was excavated, the support pressure immediately changed from p to 0, and a convergence displacement of the surrounding rock occurred. Later, the lining structure would be reset, and the support pressure increased with the convergence
displacements of the excavation surface. Finally, a new balance state was reached with support pressure s0. In order to distinguish the variables, the subscript “(i)” was employed to denote the variable for residual and peak rock masses. For example: i¼1 and 2 for the residual plastic and peak plastic rock masses, respectively. Symbol “( )” denotes that the corresponding variable is a function of radius r, and the default value of i¼ 1 and 2. Due to the axisymmetic character of the circular openings, the tangential stress ( σθ ) and radial stress ( σr ) were the major and minor stresses, respectively. Two of the most commonly used yield criteria were employed. For the peak and residual rock masses, the linear M–C criterion can be described as follows:
σ ( θi) = αiσ ( ri) + Yi
(2a)
Also, the nonlinear H–B criterion was as follows:
σ ( θi) = σ ( ri) +
mi σcσ ( ri) + siσc2
(2b)
where αi = (1 + sin φi ) /(1 − sin φi ) and Yi = 2ci cos φi /(1 − sin φi ), both of which are strength parameters; φi and ci are the friction angle and cohesion of rock mass, respectively; σc is the unaxial compressive strength of the intact rock mass; and mi and si are the strength parameters. The deformation of the geo-materials generally follows the non-associated flow rule, and a Mohr–Coulomb plastic potential function is widely used. In regards to the circular opening problem, it can be expressed as follows:
gi = σ ( θi) − βi σ ( ri)
(3)
where βi = (1 + sin ψi ) /(1 − sin ψi ), and ψi is the dilation angle of the rock mass. In regards to the small strain problem, the displacement and strain of the rock mass are related to the geometric equations. The radial and tangential strains can be expressed in terms of radial displacement as follows:
ε ( ri) = Fig. 4. Critical shear plastic strain with confining pressure.
∂u(i) ∂r
,
ε ( θi) =
u (i ) r
( i = 1,
2, 3)
(4)
where ε(ri) and ε(θi) are the radial and tangent strain, and u(i) is the
Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 86 (2016) 16–22
radial displacement. During the formation of the initial stress, the displacement occurs. However, it has little influence on the engineering problem. In this study, only the displacement induced by excavation has been mentioned. Using a generalized Hooke's law, the elastic strains induced by the excavation of the tunnel can be formulated as follows:
⎧ ⎪ ε ( ri) = ⎪ ⎨ ⎪ ε ( θi) = ⎪ ⎩
1 ⎡ (1 − μi )(σ ( ri) − p) − μi (σ ( θi) − p)⎤⎦ 2Gi ⎣ 1 ⎡ (1 − μi )(σ ( θi) − p) − μi (σ ( ri) − p)⎤⎦ 2Gi ⎣
(5)
(6)
⎧ r = R 0 σ r 0 = σ r1 = σ 0 ( ) ( ) ⎪ ⎪ ⎨ r = R1 σ ( r1) = σ ( r2), u(1) = u(2) ⎪ ⎪ ⎩ r = R2 σ ( r2) = σ ( r 3), u(2) = u(3)
du(i)
+ βi
u (i ) r
= fi (r )
(14)
(7)
σ ( ri) − σ ( θi) =0 r
(10)
By substituting Eq. (2a) into Eq. (10), and using the boundary condition of Eq. (5), the stresses can uniformly obtained for the M– C rock mass:
⎧ ⎛ ⎞αi − 1 ⎪σ = σD ⎜ r ⎟ − Di ri) i−1 ( ⎪ ⎪ ⎝ Ri − 1 ⎠ ⎨ ⎪ ⎛ r ⎞αi − 1 ⎪ σ ( θi) = αiσi D− 1⎜ − Di ⎟ ⎪ ⎝ Ri − 1 ⎠ ⎩
σ2M − C =
which,
σ2H − B = p −
(16)
⎛ ⎜1 σc ⎜⎜ 2 ⎝
⎛ m2 ⎞2 ⎟ ⎜ ⎝ 4 ⎠
+
2p − Y2 for 1 + α2
m2p σc
+ s2 −
M–C
rock
mass;
and
⎞
m2 ⎟ ⎟. 8 ⎟
Then, the solutions of Eq.
⎠
(14) can be obtained as
1 r βi
∫R
r i
⎛ R ⎞ βi r βifi (r )dr + ui ⎜ i ⎟ ⎝ r⎠
(17)
By substituting Eq. (11) into Eq. (5), and then into Eq. (15), the function of fi(r) for the M–C rock mass can be rewritten as:
where
⎡ ⎛ r ⎞αi − 1⎤ 1 ⎢ M−C ⎥ + H2Mi − C ⎜ H1i ⎟ ⎥ 2Gi ⎢⎣ ⎝ Ri − 1 ⎠ ⎦ H1Mi − C
(18)
= − (1 + βi )(1 − 2μi )(Di − p),
H2Mi − C
= − [(1 − μi − βiμi )
+αi(βi − βiμi − μi )]σiD− 1 . In the same way, by substituting Eq. (12) into Eq. (5), and then into Eq. (15), the function of fi(r) for the H–B rock mass can be rewritten as:
fi (r ) =
⎛ r ⎞ ⎛ r ⎞⎤ 1 ⎡ H− B ⎢ H1i + H2Hi − B ln⎜ ⎟ + H3Hi − Bln2⎜ ⎟⎥ 2Gi ⎣ ⎝ Ri − 1 ⎠ ⎝ Ri − 1 ⎠⎦
(19) H2Hi − B=
where = (βi − βiμi − μi )Bi + (1+βi )(1 − 2μi )(σi − 1 − p); (1 + βi )(1 − 2μi )Bi + 2(βi − βiμi − μi )Ai ; and H3Hi − B = (1 + βi )(1 − 2μi )Ai . For the M–C and H–B rock masses, the analytical solution of Eq. (14) can be rewritten in a compact way, as demonstrated below:
(11)
u (i ) r
=
1 1 ⎡ β ⎣ H1i f1i(r ) + H2i f2i(r ) + H3i f3i(r ) + 2Giui Ri i 2Gi r βi + 1 −H f ( R ) − H f ( R ) − H f ( R )⎤⎦ 1i 1i
i
2i 2i
i
3i 3i
(20)
i
where
r βi + 1 r βi + αi 1 , f2Mi − C (r ) = , 1 α − βi + 1 Ri − 1 i βi + αi r βi + 1 ⎡ ⎛ r ⎞ 1 ⎤ M−C ⎢ ln⎜ ⎥, f (r ) = 0 f2Hi − B (r ) = ⎟− βi + 1 ⎢⎣ ⎝ Ri − 1 ⎠ βi + 1 ⎥⎦ 3i f1i (r ) =
(12)
miσcσi − 1 + siσc2 .
3.3. Analytical solutions for displacements For the peak plastic and residual rock masses, the total strains consisted of elastic and plastic components. Using the plastic theory and Eq. (3), the radial and tangential plastic strains of ε(pri) and ε(pθi) can be formulated by the following equation:
in
R2 ( p − σ2) 2G 2
H1Hi − B
where σiD− 1 = σi − 1 + Di , Di = Cictgφi , and σi − 1 is the radial stress at r ¼Ri 1. In the same way, the stresses of the H–B rock mass can be formulated as follows:
⎧ r r σ = Ai ln2 + Bi ln + σi − 1 ⎪ ⎪ ( ri) Ri − 1 Ri − 1 ⎨ r 2 r ⎪σ + (2Ai + Bi )ln + Bi + σi − 1 ( θi) = Ai ln R ⎪ Ri − 1 ⎩ i−1
u2 =
fi (r ) =
Without the body force, the equilibrium equation for this axisymmetric problem can be expressed as follows:
(15)
Eq. (7) illustrates that the radial displacement should be continuous at the elastic-plastic and plastic-residual interfaces. Using the elastic thick cylinder solutions, the displacement at r ¼R2 can be formulated as the following:
u (i ) =
3.2. Analytical solutions for stresses
where Ai = miσc /4 , Bi =
Then, by substituting Eq. (4) into Eq. (13), the differential equation for the radial displacement can be expressed as the following:
fi (r ) = ε (eri) + βi ε (eθi)
where σ1 is the radial stress at r ¼R1. Additionally, both the radial stresses and displacements should be continuous at the interfaces of the elasto-peak plastic, peak plastic-residual, and residual-lining. It can be expressed by the following:
dr
(13)
where
( i = 1, 2)
γ p * = f ( σ1)
+
ε pri + βi ε pθi = 0 ( ) ( )
dr
where Gi is the shear modulus of the rock mass, and μi is the Poisson's ratio of the rock mass. As for the circular opening problem, the radial stress is the minimum principal stress. Then, the critical failure conditions at r ¼R1 can be rewritten as follows:
dσ ( ri)
19
f3Hi − B (r ) =
⎡ ⎛ r ⎞ r βi + 1 ⎢ 2⎛ r ⎞ 2 2 ln ⎜ ln⎜ ⎟− ⎟+ ⎢ βi + 1 βi + 1 ⎝ Ri − 1 ⎠ ⎝ Ri − 1 ⎠ βi + 1 ⎣
(
⎤ ⎥ . 2⎥ ⎦
)
3.4. Determination of region radii Although the explicit expressions of the stresses and displacements had been obtained, they still could not be determined for
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the stress and displacement due to the unknown residual and plastic radii. The contact conditions of Eq. (7) show that the elastic stress should also satisfy the peak yield criterion at the elastic– plastic interface. Therefore, by substituting the elastic stress components into Eq. (2a), the relationship between radii R1 and R2 for the M–C rock mass can be formulated as the following:
⎛ R ⎞α2 − 1 (α2 + 1)σ1D⎜ 2 ⎟ − 2(p + D2) = 0 ⎝ R1 ⎠
Table 1 Geometric and material property parameters.
(21)
For certain types of rock masses, the plastic bearing capacity is finite, and it depends on the minimum principal stress and rock lithology. As previously mentioned, once the shear plastic strain of the peak plastic rock mass reaches its critical value, a brittle failure occurs. By using (Eqs. (5), (6), and 20), the critical condition for the peak M–C rock mass can be formulated as:
⎛ R ⎞1 + β2
( 1 − α2)( η2 + 1)σ1D − ( 1 + β2)κ ⎜ R2 ⎟ ⎝
1⎠
− 2G2f M − C ( σ1) = 0
(22)
where η2 = (1 − μ2)(α2β2 + 1) /(α2 + β2) − μ2, κ =2(p + D2)(1 − μ2) (1 − α2) /(α2 + β2) . In regards to the H–B rock mass, the elastic stresses at r ¼R2 should also satisfy the peak plastic yield criterion, and the maximum shear plastic strain for the peak plastic rock mass should also be restricted to r ¼ R1. Then, the equations of radii R1 and R2 can be obtained, as shown below in (Eqs. (23) and 24):
A2 ln2
R2 R R R B + (A2 + B2)ln 2 + A1ln2 1 + B1 ln 1 + 2 + σ0 − p R1 R1 R0 2 R0 (23)
=0
2A2 ln
1 + β2 ⎛ ⎞ R2 ⎛ R2 ⎞ R −⎜ ⎟ ⎜ 2A2 ln 2 + B2 + A2 sin ψ2⎟ R1 ⎝ R1 ⎠ R1 ⎝ ⎠
+ A2 sin ψ2 + B2 +
G2 f H − B( σ1) 1 − μ2
=0
(24)
By substituting the stress components into (Eqs. (21)–24), the radii of R1 and R2 can theoretically be solved for the M–C and H–B rock masses, respectively. If the initial stress is lower, or the strength of the rock mass is very high, a non-residual region would be formed. In such a case, the calculated radius R1 may be less than the excavation radius R0, which is obviously incorrect. Therefore, the restrictive condition for the radii of each region should be considered, as shown below:
R2 ≥ R1 ≥ R 0
(25)
By combining (Eqs. (21)–(24) and 25), the nonlinear equations for the peak plastic and residual radius can be formulated. Due to the complex expression form, it is difficult to obtain their derivatives. In this case, it is assumed that radius R2 is known. Then, radius R1 and the requirement supporting pressure scal 0 can be calculated using (Eqs. (11) and 21) and (Eqs. (12) and 24) for the M–C and H–B rock masses, respectively. If scal 0 ¼s0, then the given radius R2 would actually be correct. However, scal 0 4s0 indicates that the given radius R2 is less than the real value, and vice versa. Also, dichotomy and secant methods can be adopted to update the radius R2.
4. Example study 4.1. Case 1: Verification for elasto-brittle-plastic rock mass As previously mentioned, the proposed analytical solution for
Scheme
Value
Radius of tunnel, R0 (m) Initial stress, p(MPa) Yong's modulus, E(GPa) Poisson's ratio, v
1 1 5 0.2
M-C rock mass c2 (MPa) φ2 (deg) c1 (MPa) φ1 (deg) ψ(deg)
0.276 35 0.055 30 0
H-B rock mass m2 s2 m1 s1 sc (MPa) ψ(deg)
0.2 1.0 10 4 0.05 1.0 10 5 50 0
the circular openings is an extension of the elasto-brittle-plastic rock mass. Also, the classic input data for the soft rock mass listed in Table 1, which was used by Ogawa and Lo,19 have been considered. For the purpose of guaranteeing the same conditions as the Ogawa and Lo research examples, the critical failure shear plastic strain γ p * was assumed as 0, and 100 (infinite large) for the elasto-brittle-plastic and elasto-perfectly-plastic rock mass, respectively. Table 2 lists the calculated results using EPBM, in accordance with the closed-form solutions. Here, γp* ¼ 1.5e 4 and 3e 4 are considered for the M-C and H-B rock masses, respectively. The influence of the supporting pressure on the surrounding rock displacement and plastic radius are shown in Figs. 6 and 7. In regards to the M–C rock mass, when the support pressure s0/p40.051, only the peak plastic region developed, and the solutions based on the EPBM are the same as those of the EPM. However, once s0/po0.051, the shear plastic strain at r¼R0 would be larger than the critical shear plastic strain γp*, at which point a brittle failure occurs, and a residual plastic region is developed. When the support pressure is completely released, then R1/R0 ¼1.76 for the EBM, and R2/R0 ¼1.17 for the EPM. Meanwhile, R1/R0 ¼1.24 and R1/ R0 ¼1.38 for the EPBM. Moreover, the dimensionless displacements 2G2u0/[R0(p-s2)] are 4.21, 1.42, and 2.31 for the former three models, respectively. In regards to the H-B rock mass, the residual region develops when s0/po0.077. When the support pressure is completely released, R1/R0 ¼1.61 for the EBM; R2/R0 ¼1.24 for the EPM; and R2/R0 ¼1.43, R1/R0 ¼1.28 for the EPBM. The dimensionless displacements of the EBM, EPM, and EPBM are 3.40, 1.68, and 2.49, respectively. Therefore, such small peak plastic bearing capacities can significantly decrease the rock mass failure region, as well as the displacement. The results denote that the dimensionless displacements and plastic radius of the EPBM were located between the EPM and EBM. In fact, most of the brittle rock mass showed a certain ideal plastic bearing capacity. A reasonable peak plastic bearing capacity can provide more suitable support parameters for the underground Table 2 Computed results of radius and dimensionless displacement (Data in parenthesis are closed-form solutions). Rock mass
R1/R0 or R2/R0
2G2u0/[R0(p s2)]
M–C rock
1.76 (1.76) 1.17(1.17) 1.61 (1.61) 1.23 (1.23)
4.21 1.42 3.40 1.68
H–B rock
(4.21) (1.42) (3.40) (1.68)
Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 86 (2016) 16–22
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Fig. 7. Influence of supporting pressure on H–B rock mass. Fig. 6. Influence of supporting pressure on M–C rock mass.
engineering. However, the critical shear plastic strain should be determined by testing. 4.2. Case 2: Application in Jinping II hydropower station Fig. 8 shows the theoretical calculated triaxial compression curves of Jinping marble, and is in accordance with the tests. Therefore, the proposed EPBM was suitable for the stability analysis. The inversion principal in-situ stresses were 48.98 MPa, 55.67 MPa, and 66.16 MPa. Also, the average value of 56.94 MPa was considered. The rock mass parameters of the M–C were obtained based on the intact rock samples. However, the rock mass original joints and excavation disturbances significantly influence the strength of the rock mass. Furthermore, it is difficult to quantify their influence degree. The H–B strength criterion is derived based on the geological strength index (GSI), as shown as follows in Eq. (26):
(
Fig. 8. Theoretical compression curves of Jinping marble by EPBM.
α
)
σ1 = σ3 + mbσcσ3 + sσc2 .
(26)
where mb, sc, s, and α are the strength parameters, and can be obtained by GSI and D:
⎛ GSI − 100 ⎞ ⎛ GSI − 100 ⎞ ⎟, s = exp⎜ ⎟, mb = mi exp⎜ ⎝ 28 − 14D ⎠ ⎝ 9 − 3D ⎠ α=
1 1 − ⎡⎣ exp( − GSI /15) − exp( − 20/3)⎤⎦. 2 6
(27)
Therefore, the H–B strength criterion effectively describes the rock mass strength based on the intact rock. Similarly, the peak
and residual strength fitting curves of the intact marble are as shown in Fig. 9. The strength parameters of the intact marble were scp ¼128.05 MPa and mbp ¼ 8.877. In fact, the fresh rock mass also contain potential cracks, not along the failure rock mass. In regards to the failure rock sample, its GSI could be obtained by the fitting curve of GSIs ¼58.4. Additionally, the failure rock mass GSIr could be approximately estimated by the GSIs and rock mass GSI, or GSIr ¼ GSIs GSI/100.20 The Jinping II hydropower station diversion tunnel was excavated by TBM. Therefore, the excavation disturbance index of D¼ 0.3 was considered. Based on the geological
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strain satisfies the failure condition, a brittle failure occurs, and then it showed macro-plastic behavior. Also, the differences between the peak and residual strength decrease with the increasing confining pressure. The critical shear plastic strain of the peak plastic rock masses showed an exponent function with the confining pressure. When the critical shear plastic strain was zero and was large enough, the proposed model turned to the EBM and EPM, respectively. The calculated results of the EPBM were found to be in accordance with the closed-form solutions of the former two models. The deteriorated Young's modulus of the residual rock mass significantly enlarged the displacements of the residual rock masses. However, it had little influence on the failure and peak plastic radius. Moreover, the proposed model can provide theoretical foundation for structural optimization and design of hard brittle rock mass.
Fig. 9. Fitting curves of H–B strength criterion for Jinping marble.
Acknowledgments This research was supported by the National Natural Science Foundation of China (Project no. 51204168), and was also supported by the Fundamental Research Funds for the Central Universities (2012QNB23).
References
Fig. 10. Failure region radius tested by sonic wave at AK þ900 section.
condition of fresh rock mass, GSI ¼ 68 was taken with a comparison with the research data of Hoek1. Additionally, D ¼1.0 was taken as the completely failed rock mass. The calculated failure radius, peak plastic radius, and air surface displacement were R1 ¼5.48 m, R2 ¼7.58 m, and u ¼2.04 cm, respectively, with equivalent area circular opening of R0 ¼ 3.84 m. The field failure radius was tested using a sonic wave method, with the failure depth of eighteen boreholes, and with an average value of 1.6 m, as shown in Fig. 10. Therefore, the calculated results were found to be in good accordance with the field tests.
5. Conclusions Taking into account the peak plastic bearing capacity of the rock masses, a simple constitutive model for the brittle rock mass was proposed based on the laboratory triaxial tests. Meanwhile, a shear plastic strain was employed to describe the brittle failure condition. Moreover, an analytical solution of the proposed model for the circular openings was derived. Finally, the examples were analyzed in order to validate the proposed solutions, and several conclusions could thereby be drawn. The peak plastic bearing capacity of the brittle rock enhances with the increase of the confining pressure. When the shear plastic
1. Hoek E, Carranza-Torres CT, Corkum B. Hoek-Brown failure criterion-2002 edition. In: Proceedings of the 5th North American Rock Mechanics Symposium and 17th Tunnelling Association of Canada Conference. Toronto; 2002, 7–10 July: 267–273. 2. Brown ET, Bray JW, Ladanyi B, Hoek E. Ground response curves for rock tunnels. J Geotech Eng. 1983;109:15–39. 3. Wang Y. Ground response of circular tunnel in poorly consolidated rock. J Geotech Eng. 1996;122:703–708. 4. Carranza-Torres C, Fairhurst C. The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek-Brown failure criterion. Int J Rock Mech Min Sci. 1999;36:777–809. 5. Sharan SK. Elastic–brittle–plastic analysis of circular openings in Hoek–Brown media. Int J Rock Mech Min Sci. 2003;40:817–824. 6. Park KH, Kim YJ. Analytical solution for a circular opening in an elastic–brittle– plastic rock. Int J Rock Mech Min Sci. 2006;43:616–622. 7. Sharan SK. Analytical solutions for stresses and displacements around a circular opening in a generalized Hoek–Brown rock. Int J Rock Mech Min Sci. 2008;40:78–85. 8. Chen R, Tonon F. Closed-form solutions for a circular tunnel in elastic–brittle– plastic ground with the original and generalized hoek-brown failure criteria. Rock Mech Rock Eng. 2011;44:169–178. 9. Wang SL, Yin S. A closed-form solution for a spherical cavity in the elastic– brittle–plastic medium. Tunn Undergr Sp Technol. 2011;26:236–241. 10. Hajiabdolmajid V, Kaiser PK, Martin CD. Modelling brittle failure of rock. Int J Rock Mech Min Sci. 2002;39:731–741. 11. Alonso E, Alejano LR, Varas F, Fdez-Manin G, Carranza-Torres C. Ground response curves for rock masses exhibiting strain-softening behavior. Int J Numer Anal Methods Geomech. 2003;27:1153–1185. 12. Zhang Q, Jiang BS, Wang SL, Ge XR, Zhang HQ. Elasto-plastic analysis of a circular opening in strain-softening rock mass. Int J Rock Mech Min Sci. 2012;50:38–46. 13. Shah KR. An elasto-plastic constitutive model for brittle-ductile transition in porous rocks. Int J Rock Mech Min Sci. 1997;34:1–13. 14. Kulhawy FH. Stress deformation properties of rock and rock discontinuities. Eng Geol. 1975;9:327–350. 15. Verman M, Singh B, Viladkar MN, Jethwa JL. Effect of tunnel depth on modulus of deformation of rock mass. Rock Mech Rock Eng. 1997;30:121–127. 16. Asef MR, Reddish DJ. The impact of confining stress on the rock mass deformation modulus. Geotechnique. 2002;52:235–241. 17. Brown ET, Bray JW, Santarelli FJ. Influence of stress-dependent elastic moduli on stresses and strains around axisymmetric boreholes. Rock Mech Rock Eng. 1989;22:189–203. 18. Zhang K. Study on constitute model andfluid-solid coupling mechanism of brittle rock [Ph.D. thesis]. Wuhan, Hubei: Institute of rock and soil mechanics; 2010. 19. Ogawa T, Lo KY. Effects of dilatancy and yield criteria on displacements around tunnels. Can Geotech J. 1987;24:100–113. 20. Dinc OS, Sonmez H, Tunusluoglu C, Kasapoglu KE. A new general empirical approach for the prediction of rock mass strengths of soft to hard rock masses. Int J Rock Mech Min Sci. 2011;48:650–665.
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Technical Note
Analytical solution for a circular opening in a rock mass obeying a three-stage stress–strain curve Qiang Zhang n, Binsong Jiang, Huajian Lv 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
art ic l e i nf o Article history: Received 17 September 2015 Received in revised form 14 March 2016 Accepted 22 March 2016 Keywords: Brittle rock mass Circular opening Yield criterion Analytical solution
1. Introduction The displacements and stresses of the surrounding rock generally influence the stability of the underground engineering. A great number of research studies have attempted to predict the evolution law by examining different methods. Among these studies, an analytical method of elasto-plastic theory was the most widely used for the axisymmetric circular openings with the linear Mohr–Coulomb (M–C) criterion, as well as the nonlinear Hoek– Brown (H–B) criterion.1 Brown et al.2 Wang,3 and Carranza–Torres and Fairhurst4 proposed analytical and numerical solutions based on elastic-perfectly-plastic models, respectively. At a later point in time, Sharan5 pointed out the defects in the woprk of 2,3, and then derived the closed-form solutions of circular openings in H–B rock masses5. Park et al.6 studied the displacements of the surrounding rock with four types of elastic strain distribution forms in the plastic region. However, only the solutions based on the generalized Hooke's law were found to be correct. Similarly, utilizing the assumption of the same elastic strain distribution in thick cylinders, Sharan derived analytical solutions for circular openings using a generalized H–B criterion.7 Furthermore, Chen and Tonon8 studied the analytical solutions of circular openings in generalized H–B rock masses. However, it was determined that the solution could n
Corresponding author. E-mail address: [email protected] (Q. Zhang).
http://dx.doi.org/10.1016/j.ijrmms.2016.03.013 1365-1609/& 2016 Elsevier Ltd. All rights reserved.
be solved with a simple numerical procedure. Wang and Yin9 derived the analytical solutions for spherical cavities excavated in elasto-brittle-plastic rock masses. Along with the analytical method, a numerical method was used for the more complicated engineering. Hajiabdolmajid et al.10 proposed a CWFS model for the failure characteristics of brittle rock masses. This model was utilized to simulate the well-documented Mine-by Experiment, which demonstrated good accordance with the in situ conditions. Also, the strain-softening analysis of the circular openings was studied for the soft rock masses using both analytical and numerical methods.11,12 It is known that the above results are mainly dependent on the constitutive model. All of the former research studies regarding brittle rock masses were almost entirely based on elasto-plastic (EPM) and elasto-brittle-plastic (EBM) models. Furthermore, the majority of the laboratory tests determined that the brittle failure characteristics usually occur when the confining pressure is low. However, with the increase in confining pressure, brittle rock masses first of all usually experience a stable perfect-plastic stage before a brittle failure occurs. Moreover, the perfect plastic (even hardening) behavior would tend to appear once the confining pressure becomes very high. This is referred to as the brittle-ductile transformation process.13 In regards to shallow underground engineering, an elasto–brittle–plastic model has the ability to effectively reflect the mechanical behavior of brittle rock masses. However, it cannot be overlooked that with increasing depth, the plastic bearing capacity is gradually enhanced, especially in
Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 86 (2016) 16–22
17
Fig. 1. Triaxial tests of brittle rock mass with various confining pressures.
deep underground engineering. Moreover, the deteriorated elastic parameters should also be considered in plastic rock masses.14–17 Therefore, this current study mainly deals with a constitutive model of brittle rock mass with consideration given to the plastic bearing capacity and deterioration of the elastic parameters. Moreover, the analytical solutions for circular openings, which were based on the EBPM proposed model, were also derived.
2. Proposed constitutive model 2.1. Triaxial tests Fig. 1 illustrates the completed strain-stress curves of marble in Jinping, which had a depth of 2550 m, and was under different confining pressures.18 It can be seen that the rock first experienced an elastic stage, and then the ideal plastic behavior occurred when it satisfied the yield function. In other words, the marble possessed a certain ideal peak plastic bearing capacity, and the plastic bearing capacity was enhanced with the increase of the confining pressure. As the deformation increased, the brittle failure occurred once the plastic strain reached a certain value. Then, the residual macro-plastic stage began. Therefore, the mechanical behavior for brittle rock masses can be represented by three stages as follows: elastic stage, peak plastic stage, and residual plastic stage, as shown in Fig. 2. Additionally, it was determined that the elastic modulus and Poisson's ratio of the peak plastic stage may be different from those of the residual stage due to cracking. It was therefore obvious that the proposed EPBM combined the EPM and EBM with the brittle failure conditions. 2.2. Mechanical parameters A great deal of previous research has determined that the Poisson's ratio has only a minimal influence on the deformation of rock masses. Therefore, it is assumed as a constant value in this current study. The calculated elastic moduli of the rock masses were 37.13 GPa, 46.33 GPa, 47.55 GPa, and 52.20 GPa for the confining pressures of 5 Mpa, 10 MPa, 20 MPa, and 40 MPa, respectively. Also, a mean value of 45.80 GPa was employed. Similarly, the Poisson's ratio was taken as 0.263. Meanwhile, the dilation angle could be easily calculated by using the maximum and minimum plastic strain, and the average values of 34.46° and 12.42° were taken for the peak plastic and residual plastic rock masses, respectively. It should be noted that the initial stage of the axial strain–stress curve with a confining pressure of 10 MPa showed a significantly nonlinear compression effect induced by the rock samples’ unevenness. Therefore, an axial compression
Fig. 2. Material behavior model for EPBM. (a) Stress–strain curve and (b) strain curve.
strain of 0.0015 was considered for this study. The peak and residual strengths determine the state of the rock mass, and it is the foundation for calculating the plastic strain. The peak and residual strengths with linear Mohr–Coulomb criterion using the simplified triaxial tests curves is shown in Fig. 3. The peak cohesion cp and friction angle φp were 35.53 MPa and 35.60°, respectively. Meanwhile, the residual cohesion cr and friction angle φr were 7.38 MPa and 42.03°, respectively. The marble showed a brittle-ductile character with the increases in the confining pressure. The residual strength equaled the peak strength when the confining pressure s3 ¼61.27 MPa. When s3 461.27 MPa, the proposed model showed a strain-hardening property. In order to ensure the correctness of the strain-hardening condition, high confining pressure tests with s3 4 61.27 MPa are recommended. 2.3. Critical failure condition As stated previously, an irreversible shear plastic strain was employed to describe the critical brittle failure condition. When the brittle failure occurred, the shear plastic strain reached its ultimate value γ p*. Theoretically speaking, when the confining pressure was large enough, the mechanical behavior of the marble
18
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Fig. 3. Fitting curves of peak and residual strength with Mohr–Coulomb criterion.
turned from brittle failure, to ductile failure. Therefore, there was a maximum limit value. In this way, γ p* can be formulated as follows:
γ p * = f ( σ¯3) = γ0p + Me N ⋅ σ¯ 3
(1)
where γ is the ultimate shear plastic strain of the peak plastic rock mass; σ3is the minimum principal pressure; and γ0p , M, and N are the fitting parameters. The evolution function for γp* could be easily obtained through triaxial compression tests. The calculated ultimate shear plastic strain, as well as the fitting curve, could be represented by γ p * = 0.061 − 0.061e−0.026 ⋅ σ¯ 3, as shown in Fig. 4.
Fig. 5. Calculated model.
p*
3. Analytical solution of circular opening 3.1. Theoretical equations Fig. 5 illustrates a circular opening of the radius R0 excavated in a homogeneous, isotropic rock mass subject to a hydrostatic pressure p. The inner surface of the opening was subject to a uniform pressure, which was supplied by the elastic lining structure. When the opening was excavated, the support pressure immediately changed from p to 0, and a convergence displacement of the surrounding rock occurred. Later, the lining structure would be reset, and the support pressure increased with the convergence
displacements of the excavation surface. Finally, a new balance state was reached with support pressure s0. In order to distinguish the variables, the subscript “(i)” was employed to denote the variable for residual and peak rock masses. For example: i¼1 and 2 for the residual plastic and peak plastic rock masses, respectively. Symbol “( )” denotes that the corresponding variable is a function of radius r, and the default value of i¼ 1 and 2. Due to the axisymmetic character of the circular openings, the tangential stress ( σθ ) and radial stress ( σr ) were the major and minor stresses, respectively. Two of the most commonly used yield criteria were employed. For the peak and residual rock masses, the linear M–C criterion can be described as follows:
σ ( θi) = αiσ ( ri) + Yi
(2a)
Also, the nonlinear H–B criterion was as follows:
σ ( θi) = σ ( ri) +
mi σcσ ( ri) + siσc2
(2b)
where αi = (1 + sin φi ) /(1 − sin φi ) and Yi = 2ci cos φi /(1 − sin φi ), both of which are strength parameters; φi and ci are the friction angle and cohesion of rock mass, respectively; σc is the unaxial compressive strength of the intact rock mass; and mi and si are the strength parameters. The deformation of the geo-materials generally follows the non-associated flow rule, and a Mohr–Coulomb plastic potential function is widely used. In regards to the circular opening problem, it can be expressed as follows:
gi = σ ( θi) − βi σ ( ri)
(3)
where βi = (1 + sin ψi ) /(1 − sin ψi ), and ψi is the dilation angle of the rock mass. In regards to the small strain problem, the displacement and strain of the rock mass are related to the geometric equations. The radial and tangential strains can be expressed in terms of radial displacement as follows:
ε ( ri) = Fig. 4. Critical shear plastic strain with confining pressure.
∂u(i) ∂r
,
ε ( θi) =
u (i ) r
( i = 1,
2, 3)
(4)
where ε(ri) and ε(θi) are the radial and tangent strain, and u(i) is the
Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 86 (2016) 16–22
radial displacement. During the formation of the initial stress, the displacement occurs. However, it has little influence on the engineering problem. In this study, only the displacement induced by excavation has been mentioned. Using a generalized Hooke's law, the elastic strains induced by the excavation of the tunnel can be formulated as follows:
⎧ ⎪ ε ( ri) = ⎪ ⎨ ⎪ ε ( θi) = ⎪ ⎩
1 ⎡ (1 − μi )(σ ( ri) − p) − μi (σ ( θi) − p)⎤⎦ 2Gi ⎣ 1 ⎡ (1 − μi )(σ ( θi) − p) − μi (σ ( ri) − p)⎤⎦ 2Gi ⎣
(5)
(6)
⎧ r = R 0 σ r 0 = σ r1 = σ 0 ( ) ( ) ⎪ ⎪ ⎨ r = R1 σ ( r1) = σ ( r2), u(1) = u(2) ⎪ ⎪ ⎩ r = R2 σ ( r2) = σ ( r 3), u(2) = u(3)
du(i)
+ βi
u (i ) r
= fi (r )
(14)
(7)
σ ( ri) − σ ( θi) =0 r
(10)
By substituting Eq. (2a) into Eq. (10), and using the boundary condition of Eq. (5), the stresses can uniformly obtained for the M– C rock mass:
⎧ ⎛ ⎞αi − 1 ⎪σ = σD ⎜ r ⎟ − Di ri) i−1 ( ⎪ ⎪ ⎝ Ri − 1 ⎠ ⎨ ⎪ ⎛ r ⎞αi − 1 ⎪ σ ( θi) = αiσi D− 1⎜ − Di ⎟ ⎪ ⎝ Ri − 1 ⎠ ⎩
σ2M − C =
which,
σ2H − B = p −
(16)
⎛ ⎜1 σc ⎜⎜ 2 ⎝
⎛ m2 ⎞2 ⎟ ⎜ ⎝ 4 ⎠
+
2p − Y2 for 1 + α2
m2p σc
+ s2 −
M–C
rock
mass;
and
⎞
m2 ⎟ ⎟. 8 ⎟
Then, the solutions of Eq.
⎠
(14) can be obtained as
1 r βi
∫R
r i
⎛ R ⎞ βi r βifi (r )dr + ui ⎜ i ⎟ ⎝ r⎠
(17)
By substituting Eq. (11) into Eq. (5), and then into Eq. (15), the function of fi(r) for the M–C rock mass can be rewritten as:
where
⎡ ⎛ r ⎞αi − 1⎤ 1 ⎢ M−C ⎥ + H2Mi − C ⎜ H1i ⎟ ⎥ 2Gi ⎢⎣ ⎝ Ri − 1 ⎠ ⎦ H1Mi − C
(18)
= − (1 + βi )(1 − 2μi )(Di − p),
H2Mi − C
= − [(1 − μi − βiμi )
+αi(βi − βiμi − μi )]σiD− 1 . In the same way, by substituting Eq. (12) into Eq. (5), and then into Eq. (15), the function of fi(r) for the H–B rock mass can be rewritten as:
fi (r ) =
⎛ r ⎞ ⎛ r ⎞⎤ 1 ⎡ H− B ⎢ H1i + H2Hi − B ln⎜ ⎟ + H3Hi − Bln2⎜ ⎟⎥ 2Gi ⎣ ⎝ Ri − 1 ⎠ ⎝ Ri − 1 ⎠⎦
(19) H2Hi − B=
where = (βi − βiμi − μi )Bi + (1+βi )(1 − 2μi )(σi − 1 − p); (1 + βi )(1 − 2μi )Bi + 2(βi − βiμi − μi )Ai ; and H3Hi − B = (1 + βi )(1 − 2μi )Ai . For the M–C and H–B rock masses, the analytical solution of Eq. (14) can be rewritten in a compact way, as demonstrated below:
(11)
u (i ) r
=
1 1 ⎡ β ⎣ H1i f1i(r ) + H2i f2i(r ) + H3i f3i(r ) + 2Giui Ri i 2Gi r βi + 1 −H f ( R ) − H f ( R ) − H f ( R )⎤⎦ 1i 1i
i
2i 2i
i
3i 3i
(20)
i
where
r βi + 1 r βi + αi 1 , f2Mi − C (r ) = , 1 α − βi + 1 Ri − 1 i βi + αi r βi + 1 ⎡ ⎛ r ⎞ 1 ⎤ M−C ⎢ ln⎜ ⎥, f (r ) = 0 f2Hi − B (r ) = ⎟− βi + 1 ⎢⎣ ⎝ Ri − 1 ⎠ βi + 1 ⎥⎦ 3i f1i (r ) =
(12)
miσcσi − 1 + siσc2 .
3.3. Analytical solutions for displacements For the peak plastic and residual rock masses, the total strains consisted of elastic and plastic components. Using the plastic theory and Eq. (3), the radial and tangential plastic strains of ε(pri) and ε(pθi) can be formulated by the following equation:
in
R2 ( p − σ2) 2G 2
H1Hi − B
where σiD− 1 = σi − 1 + Di , Di = Cictgφi , and σi − 1 is the radial stress at r ¼Ri 1. In the same way, the stresses of the H–B rock mass can be formulated as follows:
⎧ r r σ = Ai ln2 + Bi ln + σi − 1 ⎪ ⎪ ( ri) Ri − 1 Ri − 1 ⎨ r 2 r ⎪σ + (2Ai + Bi )ln + Bi + σi − 1 ( θi) = Ai ln R ⎪ Ri − 1 ⎩ i−1
u2 =
fi (r ) =
Without the body force, the equilibrium equation for this axisymmetric problem can be expressed as follows:
(15)
Eq. (7) illustrates that the radial displacement should be continuous at the elastic-plastic and plastic-residual interfaces. Using the elastic thick cylinder solutions, the displacement at r ¼R2 can be formulated as the following:
u (i ) =
3.2. Analytical solutions for stresses
where Ai = miσc /4 , Bi =
Then, by substituting Eq. (4) into Eq. (13), the differential equation for the radial displacement can be expressed as the following:
fi (r ) = ε (eri) + βi ε (eθi)
where σ1 is the radial stress at r ¼R1. Additionally, both the radial stresses and displacements should be continuous at the interfaces of the elasto-peak plastic, peak plastic-residual, and residual-lining. It can be expressed by the following:
dr
(13)
where
( i = 1, 2)
γ p * = f ( σ1)
+
ε pri + βi ε pθi = 0 ( ) ( )
dr
where Gi is the shear modulus of the rock mass, and μi is the Poisson's ratio of the rock mass. As for the circular opening problem, the radial stress is the minimum principal stress. Then, the critical failure conditions at r ¼R1 can be rewritten as follows:
dσ ( ri)
19
f3Hi − B (r ) =
⎡ ⎛ r ⎞ r βi + 1 ⎢ 2⎛ r ⎞ 2 2 ln ⎜ ln⎜ ⎟− ⎟+ ⎢ βi + 1 βi + 1 ⎝ Ri − 1 ⎠ ⎝ Ri − 1 ⎠ βi + 1 ⎣
(
⎤ ⎥ . 2⎥ ⎦
)
3.4. Determination of region radii Although the explicit expressions of the stresses and displacements had been obtained, they still could not be determined for
20
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the stress and displacement due to the unknown residual and plastic radii. The contact conditions of Eq. (7) show that the elastic stress should also satisfy the peak yield criterion at the elastic– plastic interface. Therefore, by substituting the elastic stress components into Eq. (2a), the relationship between radii R1 and R2 for the M–C rock mass can be formulated as the following:
⎛ R ⎞α2 − 1 (α2 + 1)σ1D⎜ 2 ⎟ − 2(p + D2) = 0 ⎝ R1 ⎠
Table 1 Geometric and material property parameters.
(21)
For certain types of rock masses, the plastic bearing capacity is finite, and it depends on the minimum principal stress and rock lithology. As previously mentioned, once the shear plastic strain of the peak plastic rock mass reaches its critical value, a brittle failure occurs. By using (Eqs. (5), (6), and 20), the critical condition for the peak M–C rock mass can be formulated as:
⎛ R ⎞1 + β2
( 1 − α2)( η2 + 1)σ1D − ( 1 + β2)κ ⎜ R2 ⎟ ⎝
1⎠
− 2G2f M − C ( σ1) = 0
(22)
where η2 = (1 − μ2)(α2β2 + 1) /(α2 + β2) − μ2, κ =2(p + D2)(1 − μ2) (1 − α2) /(α2 + β2) . In regards to the H–B rock mass, the elastic stresses at r ¼R2 should also satisfy the peak plastic yield criterion, and the maximum shear plastic strain for the peak plastic rock mass should also be restricted to r ¼ R1. Then, the equations of radii R1 and R2 can be obtained, as shown below in (Eqs. (23) and 24):
A2 ln2
R2 R R R B + (A2 + B2)ln 2 + A1ln2 1 + B1 ln 1 + 2 + σ0 − p R1 R1 R0 2 R0 (23)
=0
2A2 ln
1 + β2 ⎛ ⎞ R2 ⎛ R2 ⎞ R −⎜ ⎟ ⎜ 2A2 ln 2 + B2 + A2 sin ψ2⎟ R1 ⎝ R1 ⎠ R1 ⎝ ⎠
+ A2 sin ψ2 + B2 +
G2 f H − B( σ1) 1 − μ2
=0
(24)
By substituting the stress components into (Eqs. (21)–24), the radii of R1 and R2 can theoretically be solved for the M–C and H–B rock masses, respectively. If the initial stress is lower, or the strength of the rock mass is very high, a non-residual region would be formed. In such a case, the calculated radius R1 may be less than the excavation radius R0, which is obviously incorrect. Therefore, the restrictive condition for the radii of each region should be considered, as shown below:
R2 ≥ R1 ≥ R 0
(25)
By combining (Eqs. (21)–(24) and 25), the nonlinear equations for the peak plastic and residual radius can be formulated. Due to the complex expression form, it is difficult to obtain their derivatives. In this case, it is assumed that radius R2 is known. Then, radius R1 and the requirement supporting pressure scal 0 can be calculated using (Eqs. (11) and 21) and (Eqs. (12) and 24) for the M–C and H–B rock masses, respectively. If scal 0 ¼s0, then the given radius R2 would actually be correct. However, scal 0 4s0 indicates that the given radius R2 is less than the real value, and vice versa. Also, dichotomy and secant methods can be adopted to update the radius R2.
4. Example study 4.1. Case 1: Verification for elasto-brittle-plastic rock mass As previously mentioned, the proposed analytical solution for
Scheme
Value
Radius of tunnel, R0 (m) Initial stress, p(MPa) Yong's modulus, E(GPa) Poisson's ratio, v
1 1 5 0.2
M-C rock mass c2 (MPa) φ2 (deg) c1 (MPa) φ1 (deg) ψ(deg)
0.276 35 0.055 30 0
H-B rock mass m2 s2 m1 s1 sc (MPa) ψ(deg)
0.2 1.0 10 4 0.05 1.0 10 5 50 0
the circular openings is an extension of the elasto-brittle-plastic rock mass. Also, the classic input data for the soft rock mass listed in Table 1, which was used by Ogawa and Lo,19 have been considered. For the purpose of guaranteeing the same conditions as the Ogawa and Lo research examples, the critical failure shear plastic strain γ p * was assumed as 0, and 100 (infinite large) for the elasto-brittle-plastic and elasto-perfectly-plastic rock mass, respectively. Table 2 lists the calculated results using EPBM, in accordance with the closed-form solutions. Here, γp* ¼ 1.5e 4 and 3e 4 are considered for the M-C and H-B rock masses, respectively. The influence of the supporting pressure on the surrounding rock displacement and plastic radius are shown in Figs. 6 and 7. In regards to the M–C rock mass, when the support pressure s0/p40.051, only the peak plastic region developed, and the solutions based on the EPBM are the same as those of the EPM. However, once s0/po0.051, the shear plastic strain at r¼R0 would be larger than the critical shear plastic strain γp*, at which point a brittle failure occurs, and a residual plastic region is developed. When the support pressure is completely released, then R1/R0 ¼1.76 for the EBM, and R2/R0 ¼1.17 for the EPM. Meanwhile, R1/R0 ¼1.24 and R1/ R0 ¼1.38 for the EPBM. Moreover, the dimensionless displacements 2G2u0/[R0(p-s2)] are 4.21, 1.42, and 2.31 for the former three models, respectively. In regards to the H-B rock mass, the residual region develops when s0/po0.077. When the support pressure is completely released, R1/R0 ¼1.61 for the EBM; R2/R0 ¼1.24 for the EPM; and R2/R0 ¼1.43, R1/R0 ¼1.28 for the EPBM. The dimensionless displacements of the EBM, EPM, and EPBM are 3.40, 1.68, and 2.49, respectively. Therefore, such small peak plastic bearing capacities can significantly decrease the rock mass failure region, as well as the displacement. The results denote that the dimensionless displacements and plastic radius of the EPBM were located between the EPM and EBM. In fact, most of the brittle rock mass showed a certain ideal plastic bearing capacity. A reasonable peak plastic bearing capacity can provide more suitable support parameters for the underground Table 2 Computed results of radius and dimensionless displacement (Data in parenthesis are closed-form solutions). Rock mass
R1/R0 or R2/R0
2G2u0/[R0(p s2)]
M–C rock
1.76 (1.76) 1.17(1.17) 1.61 (1.61) 1.23 (1.23)
4.21 1.42 3.40 1.68
H–B rock
(4.21) (1.42) (3.40) (1.68)
Q. Zhang et al. / International Journal of Rock Mechanics & Mining Sciences 86 (2016) 16–22
21
Fig. 7. Influence of supporting pressure on H–B rock mass. Fig. 6. Influence of supporting pressure on M–C rock mass.
engineering. However, the critical shear plastic strain should be determined by testing. 4.2. Case 2: Application in Jinping II hydropower station Fig. 8 shows the theoretical calculated triaxial compression curves of Jinping marble, and is in accordance with the tests. Therefore, the proposed EPBM was suitable for the stability analysis. The inversion principal in-situ stresses were 48.98 MPa, 55.67 MPa, and 66.16 MPa. Also, the average value of 56.94 MPa was considered. The rock mass parameters of the M–C were obtained based on the intact rock samples. However, the rock mass original joints and excavation disturbances significantly influence the strength of the rock mass. Furthermore, it is difficult to quantify their influence degree. The H–B strength criterion is derived based on the geological strength index (GSI), as shown as follows in Eq. (26):
(
Fig. 8. Theoretical compression curves of Jinping marble by EPBM.
α
)
σ1 = σ3 + mbσcσ3 + sσc2 .
(26)
where mb, sc, s, and α are the strength parameters, and can be obtained by GSI and D:
⎛ GSI − 100 ⎞ ⎛ GSI − 100 ⎞ ⎟, s = exp⎜ ⎟, mb = mi exp⎜ ⎝ 28 − 14D ⎠ ⎝ 9 − 3D ⎠ α=
1 1 − ⎡⎣ exp( − GSI /15) − exp( − 20/3)⎤⎦. 2 6
(27)
Therefore, the H–B strength criterion effectively describes the rock mass strength based on the intact rock. Similarly, the peak
and residual strength fitting curves of the intact marble are as shown in Fig. 9. The strength parameters of the intact marble were scp ¼128.05 MPa and mbp ¼ 8.877. In fact, the fresh rock mass also contain potential cracks, not along the failure rock mass. In regards to the failure rock sample, its GSI could be obtained by the fitting curve of GSIs ¼58.4. Additionally, the failure rock mass GSIr could be approximately estimated by the GSIs and rock mass GSI, or GSIr ¼ GSIs GSI/100.20 The Jinping II hydropower station diversion tunnel was excavated by TBM. Therefore, the excavation disturbance index of D¼ 0.3 was considered. Based on the geological
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strain satisfies the failure condition, a brittle failure occurs, and then it showed macro-plastic behavior. Also, the differences between the peak and residual strength decrease with the increasing confining pressure. The critical shear plastic strain of the peak plastic rock masses showed an exponent function with the confining pressure. When the critical shear plastic strain was zero and was large enough, the proposed model turned to the EBM and EPM, respectively. The calculated results of the EPBM were found to be in accordance with the closed-form solutions of the former two models. The deteriorated Young's modulus of the residual rock mass significantly enlarged the displacements of the residual rock masses. However, it had little influence on the failure and peak plastic radius. Moreover, the proposed model can provide theoretical foundation for structural optimization and design of hard brittle rock mass.
Fig. 9. Fitting curves of H–B strength criterion for Jinping marble.
Acknowledgments This research was supported by the National Natural Science Foundation of China (Project no. 51204168), and was also supported by the Fundamental Research Funds for the Central Universities (2012QNB23).
References
Fig. 10. Failure region radius tested by sonic wave at AK þ900 section.
condition of fresh rock mass, GSI ¼ 68 was taken with a comparison with the research data of Hoek1. Additionally, D ¼1.0 was taken as the completely failed rock mass. The calculated failure radius, peak plastic radius, and air surface displacement were R1 ¼5.48 m, R2 ¼7.58 m, and u ¼2.04 cm, respectively, with equivalent area circular opening of R0 ¼ 3.84 m. The field failure radius was tested using a sonic wave method, with the failure depth of eighteen boreholes, and with an average value of 1.6 m, as shown in Fig. 10. Therefore, the calculated results were found to be in good accordance with the field tests.
5. Conclusions Taking into account the peak plastic bearing capacity of the rock masses, a simple constitutive model for the brittle rock mass was proposed based on the laboratory triaxial tests. Meanwhile, a shear plastic strain was employed to describe the brittle failure condition. Moreover, an analytical solution of the proposed model for the circular openings was derived. Finally, the examples were analyzed in order to validate the proposed solutions, and several conclusions could thereby be drawn. The peak plastic bearing capacity of the brittle rock enhances with the increase of the confining pressure. When the shear plastic
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