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A generalized three-dimensional failure criterion for rock masses
Journal of Rock Mechanics and Geotechnical Engineering. 2012, 4 (4): 333–343
A generalized three-dimensional failure criterion for rock masses Ashok Jaiswal*, B. K. Shrivastva Department of Mining Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, India Received 29 December 2011; received in revised form 10 May 2012; accepted 16 September 2012
Abstract: The smooth convex generalized failure function, which represents 1/6 part of envelope in the deviatoric plane, is proposed. The proposed function relies on four shape parameters (Ls, a, b and c), in which two parameters (a and b) are dependent on the others. The parameter Ls is called extension ratio. The proposed failure function could be incorporated with any two-dimensional (2D) failure criteria to make it a three-dimensional (3D) version. In this paper, a mathematical formulation for incorporation of Hoek-Brown failure criterion with the proposed function is presented. The Hoek-Brown failure criterion is the most suited 2D failure criterion for geomaterials. Two types of analyses for best-fitting solution of published true tri-axial test data were made by considering (1) constant extension ratio and (2) variable extension ratio. The shape and strength parameters for different types of rocks have been determined by best-fitting the published true tri-axial test data for both the analyses. It is observed from the best-fitting solution by considering uniform extension ratio (Ls) that shape constants have a correlation with Hoek-Brown strength parameters. Thus, only two parameters (c and m) are needed for representing the 3D failure criterion for intact rock. The statistical expression between shape and Hoek-Brown strength parameters is given. In the second analysis, when considering varying extension ratio, another parameter f is introduced. The modified extension ratio is related to f and extension ratio. The results at minimum mean misfit for all the nine rocks indicate that the range of f varies from 0.7 to 1.0. It is found that mean misfit by considering varying extension ratio is lower than that in the first analysis. But it requires three parameters. A statistical expression between f and Hoek-Brown strength parameters has been established. Though coefficient of correlation is not reasonable, we may eliminate it as an extra parameter. At the end of the paper, a methodology has also been given for its application to isotropic jointed rock mass, so that it can be implemented in a numerical code for stability analysis of jointed rock mass structures. Key words: three-dimensional (3D) failure criterion; Hoek-Brown failure criterion; true tri-axial test; deviatoric plane
1 Introduction
Various failure criteria have been proposed so far for rocks (Drucker and Prager, 1952; Mogi, 1967; Wiebols and Cook, 1968; Lade, 1977; Hoek and Brown, 1980; Matsuoka and Nakai, 1982, 1985; Ramamurthy et al., 1985; Hoek, 1994; Sheorey, 1997; Maiolino, 2005; Benz et al., 2008; You, 2009). Among them, Hoek-Brown and Mohr-Coulomb failure criteria are the most popular and commonly used criteria for stability analysis of rock structures. Mohr-Coulomb is a linear failure criterion, whereas Hoek-Brown is a nonlinear failure criterion. These are two-dimensional (2D) in nature by considering that strength is primarily dependent on the minimum principal stress only. It has been observed that the intermediate principal stress has also a significant effect on the strength. Though,
Doi: 10.3724/SP.J.1235.2012.00333 Corresponding author. Tel: +91-9450533473; E-mail: [email protected]
*
researchers do stability analysis of the rock structure, preferably, considering these 2D criteria. These criteria are simple in nature and can easily be implemented in the numerical modeling, even for elasto-plastic/strain softening analysis (Medhurst and Brown, 1998; Adhikary et al., 2002; Jaiswal and Shrivastva, 2009a, 2009b). The influence of intermediate stress on strength is considerable. Therefore, the 3D failure criterion has to be considered for more precise analysis of the structure stability. A number of 3D failure criteria are proposed, but some of them do not fulfill the Drucker’s stability postulate (Drucker, 1956). Mogi (1967) carried out laboratory investigation on true tri-axial tests on various types of rocks and proposed a 3D failure criterion. The nature of locus of failure criterion in deviatoric plane is concave, thus, it has the corner problem at the tri-axis (1, 2, 3). The same problem arises with the Wiebols-Cook criterion and its modified version also (Wiebols and Cook, 1968). Drucker and Prager (1952) proposed a failure criterion
for soil (or cohesionless material). The locus of the failure envelope in the deviatoric plane is circular. This could not be adopted for the cohesive material such as rocks or rock mass. Few researchers (Matsuoka and Nakai, 1982; Maiolino, 2005; Benz et al., 2008) proposed 3D smooth convex failure criteria. The failure envelope of true tri-axial test results of rocks in the deviatoric plane is triangular with rounded corners. They considered that the strength of the geomaterials is equal when either intermediate principal stress (2) is equal to the maximum principal stress (1) or the minimum principal stress (3). It is depicted from compression and extension tri-axial test results obtained by Mogi (2007) that strength is higher in the case of extension mode compared to that in compression mode for most of the rock types. The shape of the failure envelope in deviatoric plane may not be uniform. It was observed from the laboratory true tri-axial tests conducted on concrete blocks (Kim and Lade, 1984) that the shape of the failure envelope in the deviatoric plane varies from triangular to circular with the mean stress. Thus, the authors considered both the possibilities in the analysis, i.e. uniform and varying shape of the failure envelope in the deviatoric plane at different mean stresses. An attempt was made to propose a generalized smooth convex failure envelope in the deviatoric plane. The proposed failure function may incorporate with any 2D failure criteria. The shape of the proposed failure envelope is considered to be uniform in the first analysis and varying with the mean stress in the second analysis.
2 Mathematical formulation of smooth failure function in deviatoric plane The graphical representation of failure envelope for isotropic rock is six-fold symmetrical about its hydrostatic axis in deviatoric plane (π-plane). Shape of the failure envelope in any deviatoric plane may be uniform or vary with mean stress. The size, however, of the failure envelope in any deviatoric plane is dependent on the mean stress. The failure envelope in deviatoric plane is shown in Fig. 1 and described in the subsequent sections. 2.1 Deviatoric radius ( R ) The deviatoric radius is defined here as the length from hydrostatic axis to any point (say point P) lying
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1 A P
Rθ=0
334
B
Rθ
π/3 Rθ=π/3 O
3
2
Fig. 1 Proposed failure envelope at deviatoric plane (π-plane).
on the failure envelope in the same deviatoric plane. The deviatoric radius reduces as one moves from point A ( 2 3 ) to point B ( 1 2 ) as shown in Fig. 1. It is worth noting that both points lie in the same deviatoric plane. The deviatoric radius is expressed as
R
( m 1 ) 2 ( m 2 ) 2 ( m 3 ) 2
(1)
where m ( 1 2 3 ) / 3 is the mean stress, and R is the deviatoric radius at angle θ from axis OA. The coordinate of center point of the deviatoric plane is ( m , m , m ) . The angle from axis 1 to point P in the deviatoric plane is represented by θ. It is defined as 3( 2 3 ) tan 1 (2) 2 1 2 3 It is related to Lode angle as follows:
( Lode angle) π/6
(3)
2.2 Normalized deviatoric radius ( ) To have a similar size in any deviatoric plane, deviatoric radius for a given angle θ is divided by deviatoric radius at θ = 0. It is called a normalized deviatoric radius and defined as R (4) R 0 The maximum value of normalized deviatoric radius is one at θ = 0 (point A) and minimum is Ls at θ = /3 (point B). 2.3 Extension ratio (Ls) The extension ratio is defined as a ratio of deviatoric radius at θ = /3 and 0 in the same deviatoric plane: R (5) Ls π /3 R 0
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The range of extension ratio is 0.5 to 1.0. The shape of the failure envelope in deviatoric plane is typically triangular at Ls 0.5 and circular at 1.0. Most of the researchers have a general opinion that material strength at 2 3 and 1 2 is the same. In this case, extension ratio for Mohr-Coulomb failure is expressed as Ls
3 sin 3 sin
(6)
It is observed from Eq. (6) that Ls is constant, irrespective of the mean stress. It is dependent on internal friction angle only. Whereas, in the case of Hoek-Brown failure criterion, Ls is not uniform rather increases with the mean stress. It is discussed in detail in Section 3. 2.4 Failure function in deviatoric plane The equation of failure envelope in 1/6 part is proposed as it is six-fold symmetrical in nature. It is deduced in x-y coordinate system from point A to B: y (axb 1)c
(7)
where a, b and c are the shape constants. The curve of the above function is an invert parabolic in nature, where (0, 0) is the center of the envelope, with the conditions: a 0 , b 0 and c 0 . The function satisfies both the linear as well as circular form of envelope by considering a 0 , b 1.0 , c 1.0 and a 1.0 , b 2.0 and c 0.5, respectively. The maximum value of y is equal to one at point A (x = 0 or θ = 0). The coordinate of point B on the curve at θ = /3 should satisfy the following relations: x π /3 Ls sin(π / 6) y π /3 Ls cos(π / 6)
(8)
By inputting above values of x and y (at θ = π/3) in Eq. (7), we can get:
a
y1/ c 1 xb
dy 0 (at θ = 0 or x = 0); and dx dy (2) tan(2π/3) (at θ = /3 or x Ls sin(π/6) ) dx The first derivative of the proposed curve is as
(1)
follows: dy c(axb 1)c 1 abxb 1 (10) dx It is observed from Eq. (10) that the first condition satisfied for any value of b (except b = 1) as derivative becomes indeterminate at x = 0 and should be greater than 0 as previously discussed. Thus, b should be greater than one. The second condition is satisfied by: dy c(axb 1)c 1 abxb 1 tan(2π/3) (11) dx By inputting x and y from Eq. (8) at θ = π/3 in Eq. (11),
we have:
b{[ Ls sin(π/6)]1/ c 1} tan(2π/3) Ls cos(π/6) tan(2π/3) Ls cos(π/6) b ( c 1)/ c 1/ c Ls sin(π/6) {[ Ls sin(π/6)] 1} (12) The proposed function has four basic variables (Ls, a, b, and c). Among them, parameters a and b are related to Ls and c. The following conditions must also be satisfied to fulfill the condition of convexity at any point within d2 y the curve from point A to B: 2 0. d x The second derivative of Eq. (7) with respect to x is not easy to find out the feasible range of c with respect to Ls to satisfy Eq. (12). Thus, it was determined by simply inputting the value of c at any Ls to satisfy the above condition with the help of partial differentiation through a computer program. Fig. 2 shows the feasible range, by hatched, of c for any value of Ls. c[ Ls sin(π/6)]( c 1)/ c
4
(9a)
3 c
or [ L sin(π/6)]1/ c 1 a s [ Ls cos(π/6)]b
(9b)
The tangent at points A and B (Fig. 1) should be perpendicular to OA and OB for eliminating the corner problems. Therefore, the following two conditions should be satisfied:
2 1 0 0.5
0.6
0.7
0.8
0.9
1.0
Ls
Fig. 2 Feasible range of parameter c at any Ls by satisfying convexity of the envelope.
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Squaring both sides of Eq. (16), we have
3 Implementation of proposed function with Hoek-Brown criterion The proposed function of failure envelope in deviatoric plane could be clubbed with any 2D failure criterion to convert it into 3D criterion. Hoek-Brown failure criterion is the most suitable for most geomaterials. Therefore, Hoek-Brown failure criterion is adopted. A mathematical formulation for incorporation of Hoek-Brown failure criterion with the proposed function is given in the subsequent sections. 3.1 Hoek-Brown failure criterion The basic form of Hoek-Brown failure criterion is related to the minimum and the maximum principal stresses. It is described as m 3 1 3 c s c
a
(13)
where c is uniaxial compressive strength (UCS); m, s and a are the Hoek-Brown strength parameters. For intact rock, s and a are equal to 1 and 0.5, respectively. Strength parameters (m, s) for intact rock can be determined by best-fitting the curve between 1 and 3 for a given tri-axial test data set. In this condition, the intermediate principal stress ( 2 ) is equal to the minimum principal stress ( 3 ) , which is called compressive mode. If the maximum and the intermediate principal stresses are equal at failure, it is called extension mode. 3.2 Calculation at compressive mode (2 = 3) The mean stress in the compressive mode will be as follows: c3 c3 m c1 (14) 3 where c1 and c3 are the maximum and minimum principal stresses, respectively. The subscript ‘c’ is used for compression mode (2=3). The value of c1 can be written as function of c3 using the HoekBrown failure criterion. Thus, Eq. (14) becomes
2 c32 2 m c m c3 m2 c 9 9
2 m c m 9 c3 2
s
(18) If one knows the state of stress condition at failure (point P, Fig. 1), the mean stress and the subsequently effective minimum principal stress could be determined in the same deviatoric plane from Eqs. (2) and (18), respectively. The state of stress conditions at failure (point P) and state of effective minimum principal stress (equal to intermediate stress) and its corresponding maximum principal stress (point A) lie in the same deviatoric plane. 3.3 Calculation at extension mode (1 = 2) Similarly, the effective minimum principal stress in extension mode (1=2) is calculated. The mean stress in extension mode is e1 e3 (19) m e1 3 where the subscript ‘e’ is referred to extension mode (e1 = e2). Inputting the maximum principal stress value from Eq. (13), we have 1/ 2 m e3 e3 c s c e3 m 2 e3 3 3 1/ 2
c m e3 s 3 c
(20)
Eq. (20) can be rewritten as
m e3
c3 c3 c3 3 3
2 c m e3 s 3 c
(21)
Squaring both sides of Eq. (21), we have
1/ 2
(15)
1/ 2
c m c3 s 3 c
2 2 c2 c 2 4 m m m 9 9 2
1/ 2
c m c3 s 3 c Eq. (15) can be rewritten as
m c3
(17)
The solution of the above quadratic equation in terms of minimum principal stress ( c2 c3 ) as a function of mean stress and Hoek-Brown strength parameters is given below:
1/ 2
m c3 c3 c s c m 3
s 0
(16)
4 2 e32 2 m c m e3 m2 4 c 9 9
s 0
(22)
The solution of Eq. (22) in terms of effective confinement ( e1 e2 ) is given below:
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4 c m 9 2
2 m
e3
2
2 4 c2 4 c 2 m 4 s m m 9 9 (23) 2 Eq. (23) gives the effective minimum principal stress in extension mode ( e1 e2 ) at any mean stress. The effective minimum principal stress in compression mode, point A, and extension mode, point B, as shown in Fig. 1, can be determined from Eqs. (18) and (23), respectively, at any mean stress. The coordinate of the center point (O) is (m, m, m). All the three points A, B and O lie in the same deviatoric plane. 3.4 Calculation for extension ratio As already discussed in Section 2.3, the deviatoric radius can be determined for any state of stress condition at failure from Eq. (1). At θ = 0, the intermediate stress is equal to the minimum stress. Thus, Eq. (1) becomes
R 0
( m c1 ) 2 ( m c3 )2 ( m c3 ) 2
(24)
( m c1 ) 2 2( m c3 ) 2
(25)
or
R 0
From Eq. (14), we know c1 3 m 2 c3 . Inputting this value in Eq. (25) gives R 0
6( m c3 )
(26)
where c3 can be determined from Eq. (18). At θ = /3, the maximum and the intermediate principal stresses are equal. Thus, we have
R π /3
( m e1 )2 ( m e1 ) 2 ( m e3 ) 2
2( m e1 )2 ( m e3 ) 2 By inputting e1
(27)
3 m e3 from Eq. (19) into 2
Eq. (27), we have R π /3
3 ( m e3 ) 2
(28)
where e3 can be determined from Eq. (23). The extension ratio, Ls, is a ratio of Eq. (28) to Eq. (26). It is observed that Ls is not constant with respect to the mean stress for Hoek-Brown failure criterion, but it is a constant for the Mohr-Coulomb failure criterion. Here, it is worth noting that the
strength of the geomaterials was considered as the same when either intermediate principal stress is equal to the maximum principal stress or the minimum principal stress. However, it is observed from the laboratory tri-axial test in compressive and extension modes that strength is slightly higher in extension mode compared to compression test for the same minimum principal stress (Mogi, 2007). Thus, the extension ratio would be considered higher than the calculated one from the above expressions.
4 Best-fitting of the true tri-axial test data Nine published data sets of true tri-axial tests on different types of intact rocks (Chang and Haimson, 2000; Mogi, 2007) have been taken for best-fitting of the proposed failure function by considering HoekBrown failure criterion. The state of principal stresses data at failure can be deduced in the x-y coordinate system in deviatoric plane in normalized condition, which is given as follows: x sin (29) y cos where and can be determined in combination of Eqs. (4), (26), (18) and (2). The values of Hoek-Brown strength parameters for intact rocks (c, m) and shape constants (Ls, c) of the proposed function can be determined by best-fitting the true tri-axial data sets. Other shape constants (a, b) are the function of Ls and c. The basis of best-fitting is to minimize the mean misfit of the data sets for individual rock types. It was, of course, very difficult for bestfitting the solution. Thus, a computer program has been written to determine the best-fitting solutions at mean misfit in terms of parameters for a true tri-axial test data set of particular rock type. The program was in four-stage looping of parameters, i.e. c , m, Ls and c. The computer program runs in a loop for a particular test data set of rock type to calculate the predicted value of 1 for the given 2 and 3. The difference was calculated between the predicted and actual values of 1 and kept in the memory. The calculation was made for all the data set of particular rock type at any stage of variable parameters ( c , m, Ls and c). The average difference, i.e. the mean misfit between predicted and actual values of 1, has been calculated based on the mean-square-root concept. The same procedure has been repeated for other permutation and combination
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of parameters, i.e. c , m, Ls and c. Further, the bestfitting solution at least mean misfit has been identified among all permutation and combination of parameters. The best-fitting analysis was performed by considering both possibilities: (1) uniform extension ratio (Ls) and (2) variable extension ratio as a function of mean stress (m). 4.1 Results and analysis by considering uniform Ls The best-fitting solutions in terms of Hoek-Brown strength parameters and shape parameters of the proposed function have been determined for each type of rocks by considering various permutations and combination of strength and shape parameters. The range of c for particular Ls is taken from Fig. 2. The shape of the failure envelope in any deviatoric plane is uniform in this analysis. It indicates a reasonable correlation between Ls and c at least mean misfit. Adding the boundary values of shape parameters (i.e. at Ls = 0.5, c will be 1.0, triangular shape, and at Ls = 1.0, c will be 0.5, circular shape) to the results of the initial analysis shows that c is almost linearly dependent on Ls as described below: c 1.5 Ls (30)
log10 ( m 1) (31) 4 where is a constant observed in the range of 0.7 to 1.0. After re-analyzing the true tri-axial test data by varying parameters c, m and , most of the rock failure conditions show the least misfit at approximate value of 0.75. Thus, the statistical relationship between Ls and m becomes 3m 4 log10 4 (32) Ls 1 4 The analysis has been made by considering Ls and c as a function given in Eqs. (32) and (30), respectively. The final results at mean misfit in terms of HoekBrown strength parameters are listed in Table 1. The validity of the above expression is limited for intact rocks only. Here, it is noted that the shape of the failure envelope in any deviatoric plane is uniform (i.e. Ls is considered uniform in any deviatoric plane).
Eq. (30) also satisfies the convexity condition (Fig. 2, almost lower bound solution). The proposed failure envelope in deviatoric plane at different extension ratios considering Eq. (30) is shown in Fig. 3. The above relationship was considered for further analysis. Now, only three parameters were needed (c, m, Ls) for permutation and combination to get the best-fitting solutions.
Dolomite 294.5 8 9.55 17.8 Granite 159 55.25 25.15 23.9 Amphibolites 161.5 39.75 56.32 45 Limestone 306 4.25 22.19 14.4 Sandstone 38 24.25 8.01 6.4 Trachyte 136 6.25 8.29 9.4 Andesite 152.5 33.5 18.19 — Marble 69.5 18.25 17.34 — Monzonite 269 25.5 31.09 — Note: ‘#’ means the least mean misfit determined from the proposed failure function; ‘*’ indicates the least mean misfit as discussed by You (2009).
1
o
3 Ls = 0.7
2
Fig. 3 The proposed failure envelope at deviatoric plane for different extension ratios.
The results of this analysis have also not been discussed in the paper. It is observed that Ls has a logarithmic correlation with Hoek-Brown strength parameter m in the given form:
Ls 1
Table 1 Best-fitting solution in terms of Hoek-Brown strength parameters and mean misfit considering uniform extension ratio. Types of rocks
c (MPa)
m
Misfit# (MPa)
Misfit* (MPa)
The results have been compared with the failure criterion proposed by You (2009). It is observed that the mean misfit is slightly higher for most of the rock types. But it has corner problems at tri-axis and does not satisfy Drucker’s stability postulate. Whereas, the proposed function is a smooth convex failure criterion and also fulfills the Drucker’s stability postulate. The mean misfit for Dunham dolomite is 9.55 MPa, whereas it is 17.8 MPa when considering You’s failure criterion. The mean misfit for KTB amphibolites rock is 56.32 MPa, higher than that by You’s failure criterion. The mean misfits for other rocks are almost the same by considering the proposed and You’s failure criteria. Thus, one can choose the proposed smooth convex failure criterion by considering a uniform shape in any deviatoric plane (constant Ls). It requires only two parameters, i.e. c and m, and other parameters are interrelated with Hoek-Brown parameters. The failure envelopes of all the nine rock types in deviatoric plane are shown in Fig. 4.
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339
1
1
o
o
3
3
2
2
(a) Dunham dolomite.
(b) Inada granite.
1
1
o o
3
2
(c) KTB amphibolite. 1
2
3 (d) Solenhofen limestone.
1
o
o
3 3
2
2 (e) Shirahama sandstone. 1
(f) Mizuho trachyte. 1
o
o
3
2 (g) Manazuru andesite.
3
2 (h) Yamaguchi marble.
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1
o
3
2 (i) Orikabe monzonite.
Fig. 4 Best-fitting solution of different rock types plotted in deviatoric plane (π-plane).
The shape for Dunham dolomite, Solenhofen limestone, Mizuho trachyte rocks is almost circular as the parameter m in Hoek-Brown strength is very low in this case compared to other rock types. Other rock types show triangular shape with rounded corners because of high m value. The proposed criterion estimates that c is equal to 136 MPa, whereas it is actually 100 MPa. Thus, an ambiguity is observed in Mizuho Trachyte rock, where one data set observed in Fig. 4(f) is very far from the proposed envelope in deviatoric plane at nil confinement (c). Otherwise, most of the data lie on the failure envelope in the deviatoric plane. 4.2 Results and analysis by considering variable Lms The extension ratio varies with mean stress for Hoek-Brown failure criterion. Most of the rocks show that the strength in extension mode is higher compared to that in the compression mode for the same minimum principal stress (Mogi, 2007). Therefore, the extension ratio should be larger than the calculated one (ratio of Eq. (28) to Eq. (26)). The maximum value of Ls is one for a circular shape of failure envelope. Therefore, it should not be larger than one. Thus, a power relationship is considered as Lms Lsf (33) or f
R (34) Lms π /3 R 0 where Lms is modified extension ratio, and f is a positive constant which should be equal to or less than one. When f is equal to one, rock strength in compression and extension modes for the any minimum principal stress will be the same. When f is less than one, rock strength in extension mode will be higher compared to that in the compression mode for the same minimum principal stress. The similar approach as discussed in
the previous section for determination of least misfit has been adopted. Here, Ls has been determined from Eq. (34). The parameters c, m and f have been varied, considering Lms + c = 1.5 to find out the mean misfit. The results in terms of c, m and f constants are listed in Table 2 at least mean misfit. The mean misfit is approximately equal to the one observed by You (2009). In the case of Dunham dolomite, the mean misfit is less compared to You’s failure criterion. It is observed that the mean misfit for KTB amphibolites, when considering variable extension ratio, is considerably less, i.e. 45.44 MPa, compared to constant extension ratio, i.e. 56.32 MPa. Whereas in the case of Manazuru andesite rock, constant extension ratio gives least mean misfit compared to variable extension ratio. Table 2 Best-fitting solution in terms of Hoek-Brown strength parameters and mean misfit considering variable extension ratio. c (MPa)
m
f
Misfit# (MPa)
Misfit* (MPa)
Dolomite
327.5
6.75
0.75
9.76
17.8
Granite
195.5
40.5
0.92
24.22
23.9
Amphibolites
159.5
35.375
0.92
45.44
45
Types of rocks
Limestone
310
4.75
0.87
16.18
14.4
Sandstone
43
20.25
0.95
5.86
6.4
Trachyte
138.5
6.375
0.83
8.10
9.4
Andesite
169.5
28.25
0.90
26.37
—
71
16.5
0.86
12.37
—
273.5
24.875
0.88
35.11
—
Marble Monzonite
Note: ‘#’ means the least mean misfit determined from the proposed failure function; ‘*’ indicates the least mean misfit as discussed by You (2009).
However, in general, it is observed that nonlinear extension ratio gives less mean misfit compared to constant extension ratio. One may consider nonlinear extension ratio rather than constant. But it needs three basis parameters, i.e. c, m and f. The value of
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parameters f does not hold any standard correlation with Hoek-Brown strength parameters. However, a multiple regression analysis has been made to match the statistical expression between f and Hoek-Brown parameters. It is given below: m0.035 (35) f 0.9 0.05 c The correlation of Eq. (35) is not good (R2 = 0.57). The shape of the failure envelope in deviatoric plane is varying. Therefore, it was not plotted. The plot for all nine types of rocks at 1-2 space is plotted in Fig. 5. Most of the data approximately lie on the failure envelopes in 1-2 space. It is depicted from Fig. 5 that rock strength in extension mode is higher compared to that in the compression mode. The reason is that f is
341
less than one and it increases the modified extension ratio which in turn increases the strength in extension mode. It is also observed from Fig. 5 that percentage difference at low minimum principal stress is more than that at high minimum principal stress. The value of extension ratio increases with increasing minimum effective principal stress (Section 3). It is worth noting that both the analyses considering uniform and variable extension ratios were separate. The best-fitting solutions in terms of strength parameters of Hoek-Brown failure criterion and parameters of the proposed function in deviatoric plane on mean misfit have been determined. Therefore, the basic strength parameters for Hoek-Brown failure criterion (i.e. c and m) are different as shown in Tables 1 and 2. 1 400
1 200
1 200 1 000 1 000
25 MPa 45 MPa 65 MPa 85 MPa 105 MPa 125 MPa 145 MPa
400 200 0
0
200
400
600 800 2 (MPa)
800
400 200 0
1 000 1 200
0
200
400
800
1 400
700
1 200
600
1 000
500
800
σ0 MPa
600
30 MPa 60 MPa 100 MPa 150 MPa
400
600 800 1 000 1 200 1 400 2 (MPa) (b) Inada granite.
400 20 MPa 40 MPa 60 MPa 80 MPa 105 MPa
300 200 100
200
0
0
0
200 400
600 800 1 000 1 200 1 400 1 600 2 (MPa) (c) KTB amphibolite.
0
30
60
25
50
20
40
σ
15
5 MPa 8 MPa 15 MPa 20 MPa 30 MPa 40 MPa
10 5 0
1 (MPa)
1 (MPa)
0 MPa 2 MPa 20 MPa 38 MPa 60 MPa 77 MPa 100 MPa
600
(a) Dunham dolomite.
1 600
1 (MPa)
1 (MPa)
600
1 (MPa)
1 (MPa)
800
5
10
15 20 2 (MPa) (e) Shirahama sandstone.
25
400 600 2 (MPa) (d) Solenhofen limestone.
800
30 45 MPa 60 MPa 75 MPa 100 MPa
20
σ
10 0
0
200
30
0
10
20
30 40 2 (MPa) (f) Mizuho trachyte.
50
60
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700
1 000
600 500
1 (MPa)
1 (MPa)
800 600 20 MPa 40 MPa 70 MPa
400
12.5 MPa 25 MPa 40 MPa 55 MPa 70 MPa 85 MPa
300 200
200 0
400
100 0
200
0
400
600 800 1 000 1 200 2 (MPa) (g) Manazuru andesite. 2 000
0
100
200
300 400 500 600 2 (MPa) (h) Yamaguchi marble.
700
1 800 1 600
1 (MPa)
1 400 1 200 1 000
20 MPa 40 MPa 70 MPa 100 MPa 150 MPa 200 MPa
800 600 400 200 0 0
500
1 000 1 500 2 (MPa) (i) Orikabe monzonite.
2 000
Fig. 5 Best-fitting solution of different rock types plotted in 1-2 space. Continuous lines give calculated results for the specified minimum principal stress 3.
5 Application of the proposed function with Hoek-Brown failure criterion to jointed rock mass A methodology has been developed to apply the proposed function with Hoek-Brown failure criterion to isotropic jointed rock mass. The reduced HoekBrown strength parameter GSI (geological strength index) is proposed (Hoek, 1994): GSI 100 mm mi exp 28 14 D GSI 100 (36) sm exp 9 3D 1 1 GSI 20 am exp exp 3 2 6 15 where D is the disturbance factor. The subscripts ‘m’ and ‘i’ for Hoek-Brown strength parameters represent rock mass and intact rock, respectively. The disturbance factor depends on the degree of disturbance due to blast damage and stress relaxation
on rock mass. It varies from 0, for undisturbed in-situ rock mass, to 1, for very disturbed rock mass. The relationship between rock mass strength and the minimum principal stress (equal to the intermediate principal stress) of the jointed rock mass is correlated with resembled intact rock (considering s = 1 and a = 0.5). The UCS of the resembled intact rock can be determined as follows: c c sm (37) The Hoek-Brown strength parameter m of resembled intact rock can be determined by correlating both the curves for rock mass and resembled intact rock. The shape parameters Lms, c, a and b can be determined from Eqs. (34), (30), (9) and (12), respectively.
6 Summary and conclusions The generalized smooth convex failure function at in deviatoric plane has been proposed. It can incorporate with any 2D failure criterion. A mathematical formulation for incorporating 2D Hoek-Brown failure criterion is presented. Two hypotheses were considered in the analysis. In the first analysis, extension ratio was
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considered uniform in any deviatoric plane and it was considered to vary with mean stress in the second analysis. The results in terms of mean misfit were compared with the failure criterion proposed by You (2009). The following conclusions have been drawn from both the analyses: The shape parameter c has a linear relationship with Ls. Ls also has a strong correlation with m while considering uniform extension ratio. Thus, for representing a 3D smooth convex failure criterion, only two variables, i.e. c and m, are needed, and other parameters can be determined from the statistical relationship as mentioned in the paper. The analysis indicates that the mean misfit for all the nine types of rocks is slightly higher than that by You’s failure criterion. However, one may consider it as it requires only two parameters. The second analysis, by considering varying extension ratio, shows that parameter f does not have any correlation with other parameters. Thus, for representing a 3D smooth convex failure criterion, three parameters are required, i.e. c, m and f. The results in terms of the mean misfit are approximately equal to the one determined by You (2009). However, one can choose an average value of f = 0.85. It is concluded from both analyses that one may consider varying extension ratio rather than uniform extension ratio. The proposed function along with Hoek-Brown failure criterion could also be implemented in a numerical code for jointed rock mass analysis as the methodology given in the paper.
343 Chang C, Haimson B. True triaxial strength and deformability of the German continental deep drilling program (KTB) deep hole amphibolite. Journal of Geophysical Research, 2000, 105 (B8): 1 899–1 913. Drucker D, Prager W. Soil mechanics and plastic analysis or limit design. Quarterly of Applied Mathematics, 1952, 10 (2):157–165. Drucker D C. On uniqueness in the theory of plasticity. Quarterly of Applied Mathematics, 1956, 14 (1): 35–42. Hoek E, Brown E. Empirical strength criterion for rock masses. Journal of the Geotechnical Engineering Division, ASCE, 1980, 106 (9): 1 013– 1 035. Hoek E. Strength of rock and rock masses. ISRM News, 1994, 2 (2): 4–16. Jaiswal A, Shrivastva B K. Numerical simulation of coal pillar strength. International Journal of Rock Mechanics and Mining Sciences, 2009a, 46 (4): 779–788. Jaiswal A, Shrivastva B K. Proposed hybrid method of partial extraction. Journal of Scientific and Industrial Research, 2009b, 68 (4): 307–311. Kim K M, Lade P V. Modeling rock strength in three dimensions. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1984, 21 (1): 21–33. Lade P C. Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces. International Journal of Solids and Structures, 1977, 13 (11): 1 019–1 035. Maiolino S. Proposition of a general yield function in geomechanics. Comptes Rendus Mécanique, 2005, 333 (3): 279–284. Matsuoka H, Nakai T. A new failure criterion for soils in three dimensional stresses. In: IUTAM Conference on Deformation and Failure of Granular Materials. Rotterdam: A.A. Balkema, 1982: 253–263. Matsuoka H, Nakai T. Relationship among Tresca, Mises, Mohr-Coulomb and Matsuoka-Nakai failure criteria. Soils and Foundations, 1985, 25 (4): 123–128. Medhurst T P, Brown E T. A study of the mechanical behaviour of coal for
Acknowledgements The first author is thankful to the Department of Science and Technology, India, for financial support through its fast track project scheme (SR/FTP/ETA17-2007). References
pillar design. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1998, 35 (8): 1 087–1 105. Mogi K. Effect of the intermediate principal stress on rock failure. Journal of Geophysical Research, 1967, 72 (20): 5 117–5 131. Mogi K. Experimental rock mechanics. London, U.K.: Taylor and Frances Group, 2007. Ramamurthy T, Rao G V, Rao K S. A strength criterion for rocks. Indian Geotech. Conf., 1985, 1 (1): 59–64.
Adhikary D P, Shen B, Duncan Fama M E. A study of highwall mining panel
Sheorey P R. Empirical rock failure criteria. Rotterdam: A.A. Balkema, 1997.
stability. International Journal of Rock Mechanics and Mining Sciences,
Wiebols G, Cook N. An energy criterion for the strength of rock in Polyaxial
2002, 39 (5): 643–659. Benz T, Schwab R, Kauther R A, Vermeer P A. A Hoek-Brown criterion with intrinsic material strength factorization. International Journal of Rock Mechanics and Mining Sciences, 2008, 45 (2): 210–222.
compression. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1968, 5 (6): 529–549. You M. True triaxial strength criteria for rock. International Journal of Rock Mechanics and Mining Sciences, 2009, 46 (1): 115–127.
A generalized three-dimensional failure criterion for rock masses Ashok Jaiswal*, B. K. Shrivastva Department of Mining Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, India Received 29 December 2011; received in revised form 10 May 2012; accepted 16 September 2012
Abstract: The smooth convex generalized failure function, which represents 1/6 part of envelope in the deviatoric plane, is proposed. The proposed function relies on four shape parameters (Ls, a, b and c), in which two parameters (a and b) are dependent on the others. The parameter Ls is called extension ratio. The proposed failure function could be incorporated with any two-dimensional (2D) failure criteria to make it a three-dimensional (3D) version. In this paper, a mathematical formulation for incorporation of Hoek-Brown failure criterion with the proposed function is presented. The Hoek-Brown failure criterion is the most suited 2D failure criterion for geomaterials. Two types of analyses for best-fitting solution of published true tri-axial test data were made by considering (1) constant extension ratio and (2) variable extension ratio. The shape and strength parameters for different types of rocks have been determined by best-fitting the published true tri-axial test data for both the analyses. It is observed from the best-fitting solution by considering uniform extension ratio (Ls) that shape constants have a correlation with Hoek-Brown strength parameters. Thus, only two parameters (c and m) are needed for representing the 3D failure criterion for intact rock. The statistical expression between shape and Hoek-Brown strength parameters is given. In the second analysis, when considering varying extension ratio, another parameter f is introduced. The modified extension ratio is related to f and extension ratio. The results at minimum mean misfit for all the nine rocks indicate that the range of f varies from 0.7 to 1.0. It is found that mean misfit by considering varying extension ratio is lower than that in the first analysis. But it requires three parameters. A statistical expression between f and Hoek-Brown strength parameters has been established. Though coefficient of correlation is not reasonable, we may eliminate it as an extra parameter. At the end of the paper, a methodology has also been given for its application to isotropic jointed rock mass, so that it can be implemented in a numerical code for stability analysis of jointed rock mass structures. Key words: three-dimensional (3D) failure criterion; Hoek-Brown failure criterion; true tri-axial test; deviatoric plane
1 Introduction
Various failure criteria have been proposed so far for rocks (Drucker and Prager, 1952; Mogi, 1967; Wiebols and Cook, 1968; Lade, 1977; Hoek and Brown, 1980; Matsuoka and Nakai, 1982, 1985; Ramamurthy et al., 1985; Hoek, 1994; Sheorey, 1997; Maiolino, 2005; Benz et al., 2008; You, 2009). Among them, Hoek-Brown and Mohr-Coulomb failure criteria are the most popular and commonly used criteria for stability analysis of rock structures. Mohr-Coulomb is a linear failure criterion, whereas Hoek-Brown is a nonlinear failure criterion. These are two-dimensional (2D) in nature by considering that strength is primarily dependent on the minimum principal stress only. It has been observed that the intermediate principal stress has also a significant effect on the strength. Though,
Doi: 10.3724/SP.J.1235.2012.00333 Corresponding author. Tel: +91-9450533473; E-mail: [email protected]
*
researchers do stability analysis of the rock structure, preferably, considering these 2D criteria. These criteria are simple in nature and can easily be implemented in the numerical modeling, even for elasto-plastic/strain softening analysis (Medhurst and Brown, 1998; Adhikary et al., 2002; Jaiswal and Shrivastva, 2009a, 2009b). The influence of intermediate stress on strength is considerable. Therefore, the 3D failure criterion has to be considered for more precise analysis of the structure stability. A number of 3D failure criteria are proposed, but some of them do not fulfill the Drucker’s stability postulate (Drucker, 1956). Mogi (1967) carried out laboratory investigation on true tri-axial tests on various types of rocks and proposed a 3D failure criterion. The nature of locus of failure criterion in deviatoric plane is concave, thus, it has the corner problem at the tri-axis (1, 2, 3). The same problem arises with the Wiebols-Cook criterion and its modified version also (Wiebols and Cook, 1968). Drucker and Prager (1952) proposed a failure criterion
for soil (or cohesionless material). The locus of the failure envelope in the deviatoric plane is circular. This could not be adopted for the cohesive material such as rocks or rock mass. Few researchers (Matsuoka and Nakai, 1982; Maiolino, 2005; Benz et al., 2008) proposed 3D smooth convex failure criteria. The failure envelope of true tri-axial test results of rocks in the deviatoric plane is triangular with rounded corners. They considered that the strength of the geomaterials is equal when either intermediate principal stress (2) is equal to the maximum principal stress (1) or the minimum principal stress (3). It is depicted from compression and extension tri-axial test results obtained by Mogi (2007) that strength is higher in the case of extension mode compared to that in compression mode for most of the rock types. The shape of the failure envelope in deviatoric plane may not be uniform. It was observed from the laboratory true tri-axial tests conducted on concrete blocks (Kim and Lade, 1984) that the shape of the failure envelope in the deviatoric plane varies from triangular to circular with the mean stress. Thus, the authors considered both the possibilities in the analysis, i.e. uniform and varying shape of the failure envelope in the deviatoric plane at different mean stresses. An attempt was made to propose a generalized smooth convex failure envelope in the deviatoric plane. The proposed failure function may incorporate with any 2D failure criteria. The shape of the proposed failure envelope is considered to be uniform in the first analysis and varying with the mean stress in the second analysis.
2 Mathematical formulation of smooth failure function in deviatoric plane The graphical representation of failure envelope for isotropic rock is six-fold symmetrical about its hydrostatic axis in deviatoric plane (π-plane). Shape of the failure envelope in any deviatoric plane may be uniform or vary with mean stress. The size, however, of the failure envelope in any deviatoric plane is dependent on the mean stress. The failure envelope in deviatoric plane is shown in Fig. 1 and described in the subsequent sections. 2.1 Deviatoric radius ( R ) The deviatoric radius is defined here as the length from hydrostatic axis to any point (say point P) lying
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1 A P
Rθ=0
334
B
Rθ
π/3 Rθ=π/3 O
3
2
Fig. 1 Proposed failure envelope at deviatoric plane (π-plane).
on the failure envelope in the same deviatoric plane. The deviatoric radius reduces as one moves from point A ( 2 3 ) to point B ( 1 2 ) as shown in Fig. 1. It is worth noting that both points lie in the same deviatoric plane. The deviatoric radius is expressed as
R
( m 1 ) 2 ( m 2 ) 2 ( m 3 ) 2
(1)
where m ( 1 2 3 ) / 3 is the mean stress, and R is the deviatoric radius at angle θ from axis OA. The coordinate of center point of the deviatoric plane is ( m , m , m ) . The angle from axis 1 to point P in the deviatoric plane is represented by θ. It is defined as 3( 2 3 ) tan 1 (2) 2 1 2 3 It is related to Lode angle as follows:
( Lode angle) π/6
(3)
2.2 Normalized deviatoric radius ( ) To have a similar size in any deviatoric plane, deviatoric radius for a given angle θ is divided by deviatoric radius at θ = 0. It is called a normalized deviatoric radius and defined as R (4) R 0 The maximum value of normalized deviatoric radius is one at θ = 0 (point A) and minimum is Ls at θ = /3 (point B). 2.3 Extension ratio (Ls) The extension ratio is defined as a ratio of deviatoric radius at θ = /3 and 0 in the same deviatoric plane: R (5) Ls π /3 R 0
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335
The range of extension ratio is 0.5 to 1.0. The shape of the failure envelope in deviatoric plane is typically triangular at Ls 0.5 and circular at 1.0. Most of the researchers have a general opinion that material strength at 2 3 and 1 2 is the same. In this case, extension ratio for Mohr-Coulomb failure is expressed as Ls
3 sin 3 sin
(6)
It is observed from Eq. (6) that Ls is constant, irrespective of the mean stress. It is dependent on internal friction angle only. Whereas, in the case of Hoek-Brown failure criterion, Ls is not uniform rather increases with the mean stress. It is discussed in detail in Section 3. 2.4 Failure function in deviatoric plane The equation of failure envelope in 1/6 part is proposed as it is six-fold symmetrical in nature. It is deduced in x-y coordinate system from point A to B: y (axb 1)c
(7)
where a, b and c are the shape constants. The curve of the above function is an invert parabolic in nature, where (0, 0) is the center of the envelope, with the conditions: a 0 , b 0 and c 0 . The function satisfies both the linear as well as circular form of envelope by considering a 0 , b 1.0 , c 1.0 and a 1.0 , b 2.0 and c 0.5, respectively. The maximum value of y is equal to one at point A (x = 0 or θ = 0). The coordinate of point B on the curve at θ = /3 should satisfy the following relations: x π /3 Ls sin(π / 6) y π /3 Ls cos(π / 6)
(8)
By inputting above values of x and y (at θ = π/3) in Eq. (7), we can get:
a
y1/ c 1 xb
dy 0 (at θ = 0 or x = 0); and dx dy (2) tan(2π/3) (at θ = /3 or x Ls sin(π/6) ) dx The first derivative of the proposed curve is as
(1)
follows: dy c(axb 1)c 1 abxb 1 (10) dx It is observed from Eq. (10) that the first condition satisfied for any value of b (except b = 1) as derivative becomes indeterminate at x = 0 and should be greater than 0 as previously discussed. Thus, b should be greater than one. The second condition is satisfied by: dy c(axb 1)c 1 abxb 1 tan(2π/3) (11) dx By inputting x and y from Eq. (8) at θ = π/3 in Eq. (11),
we have:
b{[ Ls sin(π/6)]1/ c 1} tan(2π/3) Ls cos(π/6) tan(2π/3) Ls cos(π/6) b ( c 1)/ c 1/ c Ls sin(π/6) {[ Ls sin(π/6)] 1} (12) The proposed function has four basic variables (Ls, a, b, and c). Among them, parameters a and b are related to Ls and c. The following conditions must also be satisfied to fulfill the condition of convexity at any point within d2 y the curve from point A to B: 2 0. d x The second derivative of Eq. (7) with respect to x is not easy to find out the feasible range of c with respect to Ls to satisfy Eq. (12). Thus, it was determined by simply inputting the value of c at any Ls to satisfy the above condition with the help of partial differentiation through a computer program. Fig. 2 shows the feasible range, by hatched, of c for any value of Ls. c[ Ls sin(π/6)]( c 1)/ c
4
(9a)
3 c
or [ L sin(π/6)]1/ c 1 a s [ Ls cos(π/6)]b
(9b)
The tangent at points A and B (Fig. 1) should be perpendicular to OA and OB for eliminating the corner problems. Therefore, the following two conditions should be satisfied:
2 1 0 0.5
0.6
0.7
0.8
0.9
1.0
Ls
Fig. 2 Feasible range of parameter c at any Ls by satisfying convexity of the envelope.
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Squaring both sides of Eq. (16), we have
3 Implementation of proposed function with Hoek-Brown criterion The proposed function of failure envelope in deviatoric plane could be clubbed with any 2D failure criterion to convert it into 3D criterion. Hoek-Brown failure criterion is the most suitable for most geomaterials. Therefore, Hoek-Brown failure criterion is adopted. A mathematical formulation for incorporation of Hoek-Brown failure criterion with the proposed function is given in the subsequent sections. 3.1 Hoek-Brown failure criterion The basic form of Hoek-Brown failure criterion is related to the minimum and the maximum principal stresses. It is described as m 3 1 3 c s c
a
(13)
where c is uniaxial compressive strength (UCS); m, s and a are the Hoek-Brown strength parameters. For intact rock, s and a are equal to 1 and 0.5, respectively. Strength parameters (m, s) for intact rock can be determined by best-fitting the curve between 1 and 3 for a given tri-axial test data set. In this condition, the intermediate principal stress ( 2 ) is equal to the minimum principal stress ( 3 ) , which is called compressive mode. If the maximum and the intermediate principal stresses are equal at failure, it is called extension mode. 3.2 Calculation at compressive mode (2 = 3) The mean stress in the compressive mode will be as follows: c3 c3 m c1 (14) 3 where c1 and c3 are the maximum and minimum principal stresses, respectively. The subscript ‘c’ is used for compression mode (2=3). The value of c1 can be written as function of c3 using the HoekBrown failure criterion. Thus, Eq. (14) becomes
2 c32 2 m c m c3 m2 c 9 9
2 m c m 9 c3 2
s
(18) If one knows the state of stress condition at failure (point P, Fig. 1), the mean stress and the subsequently effective minimum principal stress could be determined in the same deviatoric plane from Eqs. (2) and (18), respectively. The state of stress conditions at failure (point P) and state of effective minimum principal stress (equal to intermediate stress) and its corresponding maximum principal stress (point A) lie in the same deviatoric plane. 3.3 Calculation at extension mode (1 = 2) Similarly, the effective minimum principal stress in extension mode (1=2) is calculated. The mean stress in extension mode is e1 e3 (19) m e1 3 where the subscript ‘e’ is referred to extension mode (e1 = e2). Inputting the maximum principal stress value from Eq. (13), we have 1/ 2 m e3 e3 c s c e3 m 2 e3 3 3 1/ 2
c m e3 s 3 c
(20)
Eq. (20) can be rewritten as
m e3
c3 c3 c3 3 3
2 c m e3 s 3 c
(21)
Squaring both sides of Eq. (21), we have
1/ 2
(15)
1/ 2
c m c3 s 3 c
2 2 c2 c 2 4 m m m 9 9 2
1/ 2
c m c3 s 3 c Eq. (15) can be rewritten as
m c3
(17)
The solution of the above quadratic equation in terms of minimum principal stress ( c2 c3 ) as a function of mean stress and Hoek-Brown strength parameters is given below:
1/ 2
m c3 c3 c s c m 3
s 0
(16)
4 2 e32 2 m c m e3 m2 4 c 9 9
s 0
(22)
The solution of Eq. (22) in terms of effective confinement ( e1 e2 ) is given below:
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4 c m 9 2
2 m
e3
2
2 4 c2 4 c 2 m 4 s m m 9 9 (23) 2 Eq. (23) gives the effective minimum principal stress in extension mode ( e1 e2 ) at any mean stress. The effective minimum principal stress in compression mode, point A, and extension mode, point B, as shown in Fig. 1, can be determined from Eqs. (18) and (23), respectively, at any mean stress. The coordinate of the center point (O) is (m, m, m). All the three points A, B and O lie in the same deviatoric plane. 3.4 Calculation for extension ratio As already discussed in Section 2.3, the deviatoric radius can be determined for any state of stress condition at failure from Eq. (1). At θ = 0, the intermediate stress is equal to the minimum stress. Thus, Eq. (1) becomes
R 0
( m c1 ) 2 ( m c3 )2 ( m c3 ) 2
(24)
( m c1 ) 2 2( m c3 ) 2
(25)
or
R 0
From Eq. (14), we know c1 3 m 2 c3 . Inputting this value in Eq. (25) gives R 0
6( m c3 )
(26)
where c3 can be determined from Eq. (18). At θ = /3, the maximum and the intermediate principal stresses are equal. Thus, we have
R π /3
( m e1 )2 ( m e1 ) 2 ( m e3 ) 2
2( m e1 )2 ( m e3 ) 2 By inputting e1
(27)
3 m e3 from Eq. (19) into 2
Eq. (27), we have R π /3
3 ( m e3 ) 2
(28)
where e3 can be determined from Eq. (23). The extension ratio, Ls, is a ratio of Eq. (28) to Eq. (26). It is observed that Ls is not constant with respect to the mean stress for Hoek-Brown failure criterion, but it is a constant for the Mohr-Coulomb failure criterion. Here, it is worth noting that the
strength of the geomaterials was considered as the same when either intermediate principal stress is equal to the maximum principal stress or the minimum principal stress. However, it is observed from the laboratory tri-axial test in compressive and extension modes that strength is slightly higher in extension mode compared to compression test for the same minimum principal stress (Mogi, 2007). Thus, the extension ratio would be considered higher than the calculated one from the above expressions.
4 Best-fitting of the true tri-axial test data Nine published data sets of true tri-axial tests on different types of intact rocks (Chang and Haimson, 2000; Mogi, 2007) have been taken for best-fitting of the proposed failure function by considering HoekBrown failure criterion. The state of principal stresses data at failure can be deduced in the x-y coordinate system in deviatoric plane in normalized condition, which is given as follows: x sin (29) y cos where and can be determined in combination of Eqs. (4), (26), (18) and (2). The values of Hoek-Brown strength parameters for intact rocks (c, m) and shape constants (Ls, c) of the proposed function can be determined by best-fitting the true tri-axial data sets. Other shape constants (a, b) are the function of Ls and c. The basis of best-fitting is to minimize the mean misfit of the data sets for individual rock types. It was, of course, very difficult for bestfitting the solution. Thus, a computer program has been written to determine the best-fitting solutions at mean misfit in terms of parameters for a true tri-axial test data set of particular rock type. The program was in four-stage looping of parameters, i.e. c , m, Ls and c. The computer program runs in a loop for a particular test data set of rock type to calculate the predicted value of 1 for the given 2 and 3. The difference was calculated between the predicted and actual values of 1 and kept in the memory. The calculation was made for all the data set of particular rock type at any stage of variable parameters ( c , m, Ls and c). The average difference, i.e. the mean misfit between predicted and actual values of 1, has been calculated based on the mean-square-root concept. The same procedure has been repeated for other permutation and combination
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of parameters, i.e. c , m, Ls and c. Further, the bestfitting solution at least mean misfit has been identified among all permutation and combination of parameters. The best-fitting analysis was performed by considering both possibilities: (1) uniform extension ratio (Ls) and (2) variable extension ratio as a function of mean stress (m). 4.1 Results and analysis by considering uniform Ls The best-fitting solutions in terms of Hoek-Brown strength parameters and shape parameters of the proposed function have been determined for each type of rocks by considering various permutations and combination of strength and shape parameters. The range of c for particular Ls is taken from Fig. 2. The shape of the failure envelope in any deviatoric plane is uniform in this analysis. It indicates a reasonable correlation between Ls and c at least mean misfit. Adding the boundary values of shape parameters (i.e. at Ls = 0.5, c will be 1.0, triangular shape, and at Ls = 1.0, c will be 0.5, circular shape) to the results of the initial analysis shows that c is almost linearly dependent on Ls as described below: c 1.5 Ls (30)
log10 ( m 1) (31) 4 where is a constant observed in the range of 0.7 to 1.0. After re-analyzing the true tri-axial test data by varying parameters c, m and , most of the rock failure conditions show the least misfit at approximate value of 0.75. Thus, the statistical relationship between Ls and m becomes 3m 4 log10 4 (32) Ls 1 4 The analysis has been made by considering Ls and c as a function given in Eqs. (32) and (30), respectively. The final results at mean misfit in terms of HoekBrown strength parameters are listed in Table 1. The validity of the above expression is limited for intact rocks only. Here, it is noted that the shape of the failure envelope in any deviatoric plane is uniform (i.e. Ls is considered uniform in any deviatoric plane).
Eq. (30) also satisfies the convexity condition (Fig. 2, almost lower bound solution). The proposed failure envelope in deviatoric plane at different extension ratios considering Eq. (30) is shown in Fig. 3. The above relationship was considered for further analysis. Now, only three parameters were needed (c, m, Ls) for permutation and combination to get the best-fitting solutions.
Dolomite 294.5 8 9.55 17.8 Granite 159 55.25 25.15 23.9 Amphibolites 161.5 39.75 56.32 45 Limestone 306 4.25 22.19 14.4 Sandstone 38 24.25 8.01 6.4 Trachyte 136 6.25 8.29 9.4 Andesite 152.5 33.5 18.19 — Marble 69.5 18.25 17.34 — Monzonite 269 25.5 31.09 — Note: ‘#’ means the least mean misfit determined from the proposed failure function; ‘*’ indicates the least mean misfit as discussed by You (2009).
1
o
3 Ls = 0.7
2
Fig. 3 The proposed failure envelope at deviatoric plane for different extension ratios.
The results of this analysis have also not been discussed in the paper. It is observed that Ls has a logarithmic correlation with Hoek-Brown strength parameter m in the given form:
Ls 1
Table 1 Best-fitting solution in terms of Hoek-Brown strength parameters and mean misfit considering uniform extension ratio. Types of rocks
c (MPa)
m
Misfit# (MPa)
Misfit* (MPa)
The results have been compared with the failure criterion proposed by You (2009). It is observed that the mean misfit is slightly higher for most of the rock types. But it has corner problems at tri-axis and does not satisfy Drucker’s stability postulate. Whereas, the proposed function is a smooth convex failure criterion and also fulfills the Drucker’s stability postulate. The mean misfit for Dunham dolomite is 9.55 MPa, whereas it is 17.8 MPa when considering You’s failure criterion. The mean misfit for KTB amphibolites rock is 56.32 MPa, higher than that by You’s failure criterion. The mean misfits for other rocks are almost the same by considering the proposed and You’s failure criteria. Thus, one can choose the proposed smooth convex failure criterion by considering a uniform shape in any deviatoric plane (constant Ls). It requires only two parameters, i.e. c and m, and other parameters are interrelated with Hoek-Brown parameters. The failure envelopes of all the nine rock types in deviatoric plane are shown in Fig. 4.
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339
1
1
o
o
3
3
2
2
(a) Dunham dolomite.
(b) Inada granite.
1
1
o o
3
2
(c) KTB amphibolite. 1
2
3 (d) Solenhofen limestone.
1
o
o
3 3
2
2 (e) Shirahama sandstone. 1
(f) Mizuho trachyte. 1
o
o
3
2 (g) Manazuru andesite.
3
2 (h) Yamaguchi marble.
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1
o
3
2 (i) Orikabe monzonite.
Fig. 4 Best-fitting solution of different rock types plotted in deviatoric plane (π-plane).
The shape for Dunham dolomite, Solenhofen limestone, Mizuho trachyte rocks is almost circular as the parameter m in Hoek-Brown strength is very low in this case compared to other rock types. Other rock types show triangular shape with rounded corners because of high m value. The proposed criterion estimates that c is equal to 136 MPa, whereas it is actually 100 MPa. Thus, an ambiguity is observed in Mizuho Trachyte rock, where one data set observed in Fig. 4(f) is very far from the proposed envelope in deviatoric plane at nil confinement (c). Otherwise, most of the data lie on the failure envelope in the deviatoric plane. 4.2 Results and analysis by considering variable Lms The extension ratio varies with mean stress for Hoek-Brown failure criterion. Most of the rocks show that the strength in extension mode is higher compared to that in the compression mode for the same minimum principal stress (Mogi, 2007). Therefore, the extension ratio should be larger than the calculated one (ratio of Eq. (28) to Eq. (26)). The maximum value of Ls is one for a circular shape of failure envelope. Therefore, it should not be larger than one. Thus, a power relationship is considered as Lms Lsf (33) or f
R (34) Lms π /3 R 0 where Lms is modified extension ratio, and f is a positive constant which should be equal to or less than one. When f is equal to one, rock strength in compression and extension modes for the any minimum principal stress will be the same. When f is less than one, rock strength in extension mode will be higher compared to that in the compression mode for the same minimum principal stress. The similar approach as discussed in
the previous section for determination of least misfit has been adopted. Here, Ls has been determined from Eq. (34). The parameters c, m and f have been varied, considering Lms + c = 1.5 to find out the mean misfit. The results in terms of c, m and f constants are listed in Table 2 at least mean misfit. The mean misfit is approximately equal to the one observed by You (2009). In the case of Dunham dolomite, the mean misfit is less compared to You’s failure criterion. It is observed that the mean misfit for KTB amphibolites, when considering variable extension ratio, is considerably less, i.e. 45.44 MPa, compared to constant extension ratio, i.e. 56.32 MPa. Whereas in the case of Manazuru andesite rock, constant extension ratio gives least mean misfit compared to variable extension ratio. Table 2 Best-fitting solution in terms of Hoek-Brown strength parameters and mean misfit considering variable extension ratio. c (MPa)
m
f
Misfit# (MPa)
Misfit* (MPa)
Dolomite
327.5
6.75
0.75
9.76
17.8
Granite
195.5
40.5
0.92
24.22
23.9
Amphibolites
159.5
35.375
0.92
45.44
45
Types of rocks
Limestone
310
4.75
0.87
16.18
14.4
Sandstone
43
20.25
0.95
5.86
6.4
Trachyte
138.5
6.375
0.83
8.10
9.4
Andesite
169.5
28.25
0.90
26.37
—
71
16.5
0.86
12.37
—
273.5
24.875
0.88
35.11
—
Marble Monzonite
Note: ‘#’ means the least mean misfit determined from the proposed failure function; ‘*’ indicates the least mean misfit as discussed by You (2009).
However, in general, it is observed that nonlinear extension ratio gives less mean misfit compared to constant extension ratio. One may consider nonlinear extension ratio rather than constant. But it needs three basis parameters, i.e. c, m and f. The value of
Ashok Jaiswal et al. / J Rock Mech Geotech Eng. 2012, 4 (4): 333–343
parameters f does not hold any standard correlation with Hoek-Brown strength parameters. However, a multiple regression analysis has been made to match the statistical expression between f and Hoek-Brown parameters. It is given below: m0.035 (35) f 0.9 0.05 c The correlation of Eq. (35) is not good (R2 = 0.57). The shape of the failure envelope in deviatoric plane is varying. Therefore, it was not plotted. The plot for all nine types of rocks at 1-2 space is plotted in Fig. 5. Most of the data approximately lie on the failure envelopes in 1-2 space. It is depicted from Fig. 5 that rock strength in extension mode is higher compared to that in the compression mode. The reason is that f is
341
less than one and it increases the modified extension ratio which in turn increases the strength in extension mode. It is also observed from Fig. 5 that percentage difference at low minimum principal stress is more than that at high minimum principal stress. The value of extension ratio increases with increasing minimum effective principal stress (Section 3). It is worth noting that both the analyses considering uniform and variable extension ratios were separate. The best-fitting solutions in terms of strength parameters of Hoek-Brown failure criterion and parameters of the proposed function in deviatoric plane on mean misfit have been determined. Therefore, the basic strength parameters for Hoek-Brown failure criterion (i.e. c and m) are different as shown in Tables 1 and 2. 1 400
1 200
1 200 1 000 1 000
25 MPa 45 MPa 65 MPa 85 MPa 105 MPa 125 MPa 145 MPa
400 200 0
0
200
400
600 800 2 (MPa)
800
400 200 0
1 000 1 200
0
200
400
800
1 400
700
1 200
600
1 000
500
800
σ0 MPa
600
30 MPa 60 MPa 100 MPa 150 MPa
400
600 800 1 000 1 200 1 400 2 (MPa) (b) Inada granite.
400 20 MPa 40 MPa 60 MPa 80 MPa 105 MPa
300 200 100
200
0
0
0
200 400
600 800 1 000 1 200 1 400 1 600 2 (MPa) (c) KTB amphibolite.
0
30
60
25
50
20
40
σ
15
5 MPa 8 MPa 15 MPa 20 MPa 30 MPa 40 MPa
10 5 0
1 (MPa)
1 (MPa)
0 MPa 2 MPa 20 MPa 38 MPa 60 MPa 77 MPa 100 MPa
600
(a) Dunham dolomite.
1 600
1 (MPa)
1 (MPa)
600
1 (MPa)
1 (MPa)
800
5
10
15 20 2 (MPa) (e) Shirahama sandstone.
25
400 600 2 (MPa) (d) Solenhofen limestone.
800
30 45 MPa 60 MPa 75 MPa 100 MPa
20
σ
10 0
0
200
30
0
10
20
30 40 2 (MPa) (f) Mizuho trachyte.
50
60
342
Ashok Jaiswal et al. / J Rock Mech Geotech Eng. 2012, 4 (4): 333–343 1 200
700
1 000
600 500
1 (MPa)
1 (MPa)
800 600 20 MPa 40 MPa 70 MPa
400
12.5 MPa 25 MPa 40 MPa 55 MPa 70 MPa 85 MPa
300 200
200 0
400
100 0
200
0
400
600 800 1 000 1 200 2 (MPa) (g) Manazuru andesite. 2 000
0
100
200
300 400 500 600 2 (MPa) (h) Yamaguchi marble.
700
1 800 1 600
1 (MPa)
1 400 1 200 1 000
20 MPa 40 MPa 70 MPa 100 MPa 150 MPa 200 MPa
800 600 400 200 0 0
500
1 000 1 500 2 (MPa) (i) Orikabe monzonite.
2 000
Fig. 5 Best-fitting solution of different rock types plotted in 1-2 space. Continuous lines give calculated results for the specified minimum principal stress 3.
5 Application of the proposed function with Hoek-Brown failure criterion to jointed rock mass A methodology has been developed to apply the proposed function with Hoek-Brown failure criterion to isotropic jointed rock mass. The reduced HoekBrown strength parameter GSI (geological strength index) is proposed (Hoek, 1994): GSI 100 mm mi exp 28 14 D GSI 100 (36) sm exp 9 3D 1 1 GSI 20 am exp exp 3 2 6 15 where D is the disturbance factor. The subscripts ‘m’ and ‘i’ for Hoek-Brown strength parameters represent rock mass and intact rock, respectively. The disturbance factor depends on the degree of disturbance due to blast damage and stress relaxation
on rock mass. It varies from 0, for undisturbed in-situ rock mass, to 1, for very disturbed rock mass. The relationship between rock mass strength and the minimum principal stress (equal to the intermediate principal stress) of the jointed rock mass is correlated with resembled intact rock (considering s = 1 and a = 0.5). The UCS of the resembled intact rock can be determined as follows: c c sm (37) The Hoek-Brown strength parameter m of resembled intact rock can be determined by correlating both the curves for rock mass and resembled intact rock. The shape parameters Lms, c, a and b can be determined from Eqs. (34), (30), (9) and (12), respectively.
6 Summary and conclusions The generalized smooth convex failure function at in deviatoric plane has been proposed. It can incorporate with any 2D failure criterion. A mathematical formulation for incorporating 2D Hoek-Brown failure criterion is presented. Two hypotheses were considered in the analysis. In the first analysis, extension ratio was
Ashok Jaiswal et al. / J Rock Mech Geotech Eng. 2012, 4 (4): 333–343
considered uniform in any deviatoric plane and it was considered to vary with mean stress in the second analysis. The results in terms of mean misfit were compared with the failure criterion proposed by You (2009). The following conclusions have been drawn from both the analyses: The shape parameter c has a linear relationship with Ls. Ls also has a strong correlation with m while considering uniform extension ratio. Thus, for representing a 3D smooth convex failure criterion, only two variables, i.e. c and m, are needed, and other parameters can be determined from the statistical relationship as mentioned in the paper. The analysis indicates that the mean misfit for all the nine types of rocks is slightly higher than that by You’s failure criterion. However, one may consider it as it requires only two parameters. The second analysis, by considering varying extension ratio, shows that parameter f does not have any correlation with other parameters. Thus, for representing a 3D smooth convex failure criterion, three parameters are required, i.e. c, m and f. The results in terms of the mean misfit are approximately equal to the one determined by You (2009). However, one can choose an average value of f = 0.85. It is concluded from both analyses that one may consider varying extension ratio rather than uniform extension ratio. The proposed function along with Hoek-Brown failure criterion could also be implemented in a numerical code for jointed rock mass analysis as the methodology given in the paper.
343 Chang C, Haimson B. True triaxial strength and deformability of the German continental deep drilling program (KTB) deep hole amphibolite. Journal of Geophysical Research, 2000, 105 (B8): 1 899–1 913. Drucker D, Prager W. Soil mechanics and plastic analysis or limit design. Quarterly of Applied Mathematics, 1952, 10 (2):157–165. Drucker D C. On uniqueness in the theory of plasticity. Quarterly of Applied Mathematics, 1956, 14 (1): 35–42. Hoek E, Brown E. Empirical strength criterion for rock masses. Journal of the Geotechnical Engineering Division, ASCE, 1980, 106 (9): 1 013– 1 035. Hoek E. Strength of rock and rock masses. ISRM News, 1994, 2 (2): 4–16. Jaiswal A, Shrivastva B K. Numerical simulation of coal pillar strength. International Journal of Rock Mechanics and Mining Sciences, 2009a, 46 (4): 779–788. Jaiswal A, Shrivastva B K. Proposed hybrid method of partial extraction. Journal of Scientific and Industrial Research, 2009b, 68 (4): 307–311. Kim K M, Lade P V. Modeling rock strength in three dimensions. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1984, 21 (1): 21–33. Lade P C. Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces. International Journal of Solids and Structures, 1977, 13 (11): 1 019–1 035. Maiolino S. Proposition of a general yield function in geomechanics. Comptes Rendus Mécanique, 2005, 333 (3): 279–284. Matsuoka H, Nakai T. A new failure criterion for soils in three dimensional stresses. In: IUTAM Conference on Deformation and Failure of Granular Materials. Rotterdam: A.A. Balkema, 1982: 253–263. Matsuoka H, Nakai T. Relationship among Tresca, Mises, Mohr-Coulomb and Matsuoka-Nakai failure criteria. Soils and Foundations, 1985, 25 (4): 123–128. Medhurst T P, Brown E T. A study of the mechanical behaviour of coal for
Acknowledgements The first author is thankful to the Department of Science and Technology, India, for financial support through its fast track project scheme (SR/FTP/ETA17-2007). References
pillar design. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1998, 35 (8): 1 087–1 105. Mogi K. Effect of the intermediate principal stress on rock failure. Journal of Geophysical Research, 1967, 72 (20): 5 117–5 131. Mogi K. Experimental rock mechanics. London, U.K.: Taylor and Frances Group, 2007. Ramamurthy T, Rao G V, Rao K S. A strength criterion for rocks. Indian Geotech. Conf., 1985, 1 (1): 59–64.
Adhikary D P, Shen B, Duncan Fama M E. A study of highwall mining panel
Sheorey P R. Empirical rock failure criteria. Rotterdam: A.A. Balkema, 1997.
stability. International Journal of Rock Mechanics and Mining Sciences,
Wiebols G, Cook N. An energy criterion for the strength of rock in Polyaxial
2002, 39 (5): 643–659. Benz T, Schwab R, Kauther R A, Vermeer P A. A Hoek-Brown criterion with intrinsic material strength factorization. International Journal of Rock Mechanics and Mining Sciences, 2008, 45 (2): 210–222.
compression. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 1968, 5 (6): 529–549. You M. True triaxial strength criteria for rock. International Journal of Rock Mechanics and Mining Sciences, 2009, 46 (1): 115–127.